CBSE class 10 chapter 2 polynomials pyqs and practice questions with solutions

 

 

Polynomials Level 1.

Which of the following is the numerical coefficient of x2y2 is?

1

2

x2

y2

a

Numerical coefficient is any constant term that is in front of one or more variables in a mathematical expression

Since we know that, numerical coefficient is any constant term that is in front of one or more variables in a mathematical expression  The numerical coefficient of x2 is 1  Similarly,  the numerical coefficient of y2 is 1. Hence, The numerical coefficient of x2y2 is 1

Coefficients of a Polynomial

pqr is what type of polynomial?

Binomial

Monomial

Trinomial

None of above

b

A mathematical expression consists of a single term is known as monomial.

A mathematical expression consists of a single term is known as monomial. In monomial there is only one term and pqr is having only one term. Hence, pqr is a monomial

Types of Polynomials (Constant, Linear, Quadratic, Cubic, etc.)

x2 – 4x + 5 is a polynomial in

One variable

Two variables

Three variables

None of above

a

Note down the number of variables in the mathematical expression

Given, x2 – 4x + 5 The above expression can be written as x2 – 4x1 + 5x0. we can see that x is the only variable having powers as whole numbers, 2, 1 and 0. Hence, x2 – 4x + 5 is a polynomial in one variable

Introduction to Polynomials

The expression 4xy + 7 is in

One variable

Two variables

Three variables

None of above

b

Note down the number of variables in the mathematical expression

Given, 4xy + 7 we can see that there are two variables x and y in the expression 4xy + 7 Hence, The expression 4xy + 7 is in two variables

Introduction to Polynomials

A binomial of degree 20 in the following is

20x + 1

x20 + 1

x20 + 1

x + 20

b

Basic knowledge of polynomial

A polynomial having two terms and the highest degree 20 is called a binomial of degree 20. Hence, x20 + 1 is the correct answer

Degree of a Polynomial

The coefficient of x2 in 3x3 + x2 + 2x – 3 is
0
1
2
3
b
A numerical or constant quantity placed before and multiplying the variable in an algebric expression
Given, 3x3 + x2 + 2x – 3 Hence, The coefficient of x2 in 3x3 + x2 + 2x – 3 is 1
Coefficients of a Polynomial

√3 is a polynomial of degree

2

0

1

1/2

b

Basic knowledge of polynomial

Given, √3 √3 = √3x0 Hence, Degree is 0.

Degree of a Polynomial

Find the value of k, if x2 + kx + 6 = (x + 3)(x + 2) for all k
1
-1
5
3
c
Simply the expression and then equate both the expression to find the value of k
Given, x2 + kx + 6 = (x + 3)(x + 2) (x + 3)(x + 2) = x2 + 2x + 3x + 6 = x2 + 5x + 6 Now, x2 + kx + 6 = x2 + 5x + 6 k = 5 Hence, The value of k is 5
Introduction to Polynomials

What is the degree of a zero polynomial?

2

1

Any natural number

0

d

General polynomial p(x) of nth degree can be written as: p(x) = a0 + a1x + a2x2+ …..+ anx3, where ais are contants and an ≠0

The degree of the zero polynomial is undefined. Since we know that the general polynomial p (x) of nth degree can be written as follows: p(x) = a0 + a1x + a2x2+ …..+ anx3, where ais are contants and an ≠0. So, zero polynomial is written as: 0 = 0 + 0x + 0x2 + 0x3+….+ 0xn. Therefore, the highest power of x is zero in a zero polynomial, which implies that its degree is also 0. Hence, The degree of a zero polynomial is 0

Degree of a Polynomial

The value of 992 – 982 is
1
197
157
187
b
Use a2 – b2 = (a - b)(a + b)
Given, 992 – 982 992 – 982 = (99 + 98)(99 - 98) = 197 × 1 = 197 Hence, The value of 992 – 982 is 197
Introduction to Polynomials

The coefficient of xy in 3x2y2 is

xy

-xy

3xy

2xy

c

Basic knowledge of polynomial

Given, 3x2y2 3x2y2 = (xy) (3xy) Hence, Coefficient of xy in 3x2y2 is 3xy

Coefficients of a Polynomial

The degree of the polynomial, x2 – 4x + 3
1
2
3
4
b
Degree is the highest power of the variable in any polynomial.
Degree is the highest power of the variable in any polynomial. Hence, The degree of the polynomial, x2 – 4x + 3 is 2 as the highest power is 2.
Degree of a Polynomial

If x + 1/ x = 4, then the value of x2 + 1/x2 is,
8
14
16
20
b
Simplify the expression
Given, x + 1/ x = 4 Hence, The  value of x2 + 1/x2 is 14
Factorization of Polynomials

Which of the following is a binomial?

3ab

3x  + 2y

2x + 5y - 3

a + b + c + 5

b

Binomial consists of two terms

We know that, binomial consists of two terms Hence, 3x + 2y is a binomial term

Introduction to Polynomials

The value of 155 mod 9 is
0
1
2
3
c
Use modulo method
When we divide 155 by 9, we get remainder as 2. Hence, The value of 155 mod 9 is 2
Introduction to Polynomials

A polynomial of degree ____ is called a cubic polynomial.
1
2
3
0
c
The highest power of cubic polynomial is 3.
The general form of cubic polynomial is p(x) = ax3 + bx2 + cx + d, where x is variable and a, b, c, d are constants. Hence, A polynomial is degree 3 is called a cubic polynomial.
Degree of a Polynomial

The degree of the polynomial 2x – 1?

0

½

-1

1

d

For a polynomial the degree is the value of highest power of the variable

Given, 2x - 1 For a polynomial the degree is the value of highest power of the variable. In the given polynomial, the highest power of variable x is 1, since the degree is 1.  Hence, The degree of the polynomial 2x - 1 is 1

Degree of a Polynomial

p(x) = 25 is a ___polynomial.

Linear

Constant

Cubic

Quadratic

b

The general form of a constant polynomial is p(x) = c with constant c.

Since, p(x) = 25 is a polynomial with constant term 25 and no variable is there.  Hence, p(x) = 25 is a constant polynomial.

Types of Polynomials (Constant, Linear, Quadratic, Cubic, etc.)

Which of the following is a linear expression:

X2 + 1

Y + y2

2

1 + x

d

When an algebric expression consist of only one variable whose power is 1, then it is known as linear expression.

When an algebric expression consist of only one variable whose power is 1, then it is known as linear expression. Thus, 1 + x is the only expression in which single variable is given with power 1.  Hence, 1 + x is a linear expression

Introduction to Polynomials

If f(x) = 8, then f(x) is called

Constant polynomials

Linear polynomials

Cubic polynomials

Quadratic polynomials

a

The constant polynomials are the polynomials with no variable

If f(x) = 8, then f(x) is called constant polynomials  Hence, The answer is constant polynomial

Introduction to Polynomials

The degree of polynomial is x + 2 is:
2
1
3
4
b
Highest power of the variable is called the degree of the polynomial
Apply the property of polynomial. We have, x + 2 We know that the highest power of the variable is called the degree of the polynomial. Here, the highest power of the variable x is 1. Hence, The degree of polynomial is x + 2 is 1
Degree of a Polynomial

A polynomial whose sum and product of zeroes are -4 and 3 is

0

x2 – 4x + 3

x2 + 4x + 3

x2 - 4x -3

c

P(x) = x2 – ( sum the zeroes)x + ( product of zeroes)

Given, sum of zeroes is -4  And, product of zeroes is 3 Since we know that, p(x) = x2 – ( sum the zeroes)x + ( product of zeroes) x2 – (-4)x + 3 = x2 + 4x + 3 Hence, The polynomial is x2 + 4x + 3

Relationship between Zeroes and Coefficients of a Polynomial

The number of zeroes of a cubic polynomial is

3

At most 3

8

At least 3

b

Highest power of the variable in cubic polynomial is 3

The number of zeroes of a cubic polynomial is at most 3 because the highest power of the variable in cubic polynomial is 3 Therefore, ax3 + bx2 + cx + d is the cubic polynomial. Hence, The number of zeroes of a cubic polynomial is at most 3

Relationship between Zeroes and Coefficients of a Polynomial

The maximum number of zeroes that a polynomial of degree 3 can have is.
0
1
2
3
d
The number of zeroes of a polynomial is equal to the degree of that polynomial
The maximum number of zeroes that a polynomial of degree 3 can have is three because the number of zeroes of a polynomial is equal to the degree of that polynomial. Hence, The maximum number of zeroes that a polynomial of degree 3 can have is 3
Relationship between Zeroes and Coefficients of a Polynomial

8 is a polynomial of degree?

1

½

8

0

d

Degree of a polynomial is the highest power of the variable in the polynomial

Since we know that degree of a polynomial is the highest power of the variable in the polynomial. (since 8x0 = 8) Hence, degree of 8 is 0

Degree of a Polynomial

Find the degree of the given algebraic expression ax2 + bx + c.
0
1
2
3
c
Degree is the term with the greatest exponent
Since we know that, the degree is the term with the greatest exponent. Since the highest exponent is 2, therefore, the degree of ax2 + bx + c is 2. Hence, The degree of the algebraic expression ax2 + bx + c is 2.
Degree of a Polynomial

What is the degree of the given monomial 7y?
0
1
2
7
b
Degree is the term with the greatest exponent
We know that the degree is the term with the greatest exponent and, to find the degree of a monomial with more than one variable for the same term, just add the exponents for each variable to get the degree. Here, the given monomial 7y has one variable y where the power of y is 1.  Therefore, degree of the monomial 7y is 1 (the exponent of the variable) Hence, The degree of the monomial 7y is 1.
Degree of a Polynomial

The polynomial 3x – 2 is a?

Linear polynomial

Quadratic polynomial

Cubic polynomial

None of these

a

There are no powers greater than 1 in the given polynomial

We can clearly see that there are no powers greater than 1  Hence, The given polynomial is a linear polynomial.

Types of Polynomials (Constant, Linear, Quadratic, Cubic, etc.)

√2 is a polynomial of degree?

2

0

1

½

b

√2 is a constant term

Given, √2 We can write √2 as √2 × x0 Hence, Degree is 0.

Degree of a Polynomial

Which type of polynomial is t2?

Cubic polynomial

Quadratic polynomial

Linear polynomial

None of these

b

The degree of the given expression is 2

Given, t2  The degree of the given expression is 2.  Hence, It is a quadratic polynomial.

Types of Polynomials (Constant, Linear, Quadratic, Cubic, etc.)

Which type of polynomial are (2 + x)?

Linear polynomial

Cubic polynomial

Quadratic polynomial

None of above

a

A polynomial with degree 1 is called a linear polynomial

Given, (2 + x) The degree of the equation is 1.  A polynomial with degree 1 is called a linear polynomial. Hence, 2 + x is  a linear polynomial

Types of Polynomials (Constant, Linear, Quadratic, Cubic, etc.)

Which is a binomial of degree 20?

x20 + 1

x19 + 1

20x + 1

20xy + 1

a

If an expression contains two unlike terms, then it is called as a binomial.

Given, degree of binomial is 20  If an expression contains two unlike terms, then it is called as a binomial. For binomial of degree 20, the highest power of variable should be 20. Hence, x20 + 1 is a binomial of degree 20.

Degree of a Polynomial

Which type of polynomial is 3x3?

Cubic polynomial

Linear polynomial

Square polynomial

None of these

a

In the given polynomial, the highest power is 3

Given, 3x3 In the given polynomial, the highest power is 3 Hence, The given polynomial is a cubic polynomial as the highest power of x is 3.

Types of Polynomials (Constant, Linear, Quadratic, Cubic, etc.)

Which type of polynomial is 5t - √7?

Linear polynomial

Cubic polynomial

Quadratic polynomial

None of these

a

A polynomial of degree 1 is called a linear polynomial

Given, 5t - √7 is a polynomial in variable t and the highest power of variable t is 1. A polynomial of degree 1 is called a linear polynomial. Hence, 5t - √7 is a linear polynomial.

Types of Polynomials (Constant, Linear, Quadratic, Cubic, etc.)

A polynomial of degree _____________ is called a quadratic polynomial.
0
1
2
3
c
An equation involving a quadratic polynomial is called a quadratic equation
An equation involving a quadratic polynomial is called a quadratic equation. A quadratic equation is an equation that can be written in the form ax2 + bx + c = 0, where a ≠ 0. This form is called the standard form of a quadratic equation. Quadratic equation can have two, one or no real number solution. Hence, A polynomial of degree 2 is called a quadratic polynomial.
Degree of a Polynomial

The degree of a constant polynomial is
0
1
2
3
a
The degree of a constant polynomial is 0. We know that the degree of a constant polynomial is 0. For example: 5 = 5 × 1 = 5 × x0 Hence, The degree of a constant polynomial is 0
Degree of a Polynomial

By division algorithm of polynomial, p(x) =

g(x) × q(x) + r(x)

g(x) × q(x) - r(x)

g(x) × q(x) × r(x)

g(x) + q(x) + r(x)

a

Basic knowledge of polynomials

By division algorithm of polynomial, p(x) = g(x) × q(x) + r(x)  Hence, The correct answer is g(x) × q(x) + r(x)

Division Algorithm for Polynomials

The product of the zeroes of the cubic polynomial ax3 + bx2 + cx + d is

-b/a

c/a

-d/a

-c/a

c

Basic knowledge of polynomials

The product of the zeroes of the cubic polynomial ax3 + bx2 + cx + d is –d/a. Hence, The answer is -d/a

Relationship between Zeroes and Coefficients of a Polynomial

The zero of the polynomial p(x) = ax + b is

-b/a

b/a

-a/b

a/b

a

Put p(x) = 0

Given, polynomial p(x) = ax + b Put p(x) = 0 ax + b = 0 ax = -b x = -b/a Hence, The zero of the polynomial p(x) = a + b is -b/a

Relationship between Zeroes and Coefficients of a Polynomial

If  ‘α’ and ‘β’ are the zeroes of a quadratic polynomial ax2 + bx + c, then α + β =

c/a

b/a

-b/a

-c/a

c

Sum of the zeroes of a quadratic polynomial, ax2 + bx + c = -(coefficient of x)/ coefficient of x2

Given,  ‘α’ and ‘β’ are the zeroes of a quadratic polynomial ax2 + bx + c Since sum of the zeroes of a quadratic polynomial ax2 + bx + c = -(coefficient of x)/ coefficient of x2 Then, α +β = -b/a Hence, If  ‘α’ and ‘β’ are the zeroes of a quadratic polynomial ax2 + bx + c, then α + β is -b/a

Relationship between Zeroes and Coefficients of a Polynomial

The zeroes of the quadratic polynomial x2 + 9x + 20 are

-4 and 5

-4 and -5

4 and 5

4 and -5

b

Solve by splitting the middle term

Given, (x2 + 9x + 20) Splitting the middle term, we get X2 + 5x + 4x + 20 = 0 = x(x + 5) + 4(x+ 5) = 0 = (x+5) (x+ 4) = 0 Since, x + 5 = 0 and x + 4 = 0 So, x = -5 and x = -4 Hence, The zeroes of the quadratic polynomial x2 + 9x + 20 are -5 and -4

Relationship between Zeroes and Coefficients of a Polynomial

The roots of the equation 𝑥2 + 𝑥 − 𝑝(𝑝 + 1) = 0, where 𝑝 is a constant, are

𝑝, 𝑝 + 1

−𝑝, 𝑝 + 1

𝑝, −(𝑝 + 1)

−𝑝, −(𝑝 + 1)

c

simplify the equation

Given equation  𝑥2 + 𝑥 − 𝑝(𝑝 + 1) = 0 solving further x2 + x - p2 + p = 0 x2 - p2 + x - p = 0 (x + p) (x - p) + x - p = 0 (x - p) [(x + p) + 1] = 0 so x = p or x = -( p + 1) Hence, The roots of the equation 𝑥2 + 𝑥 − 𝑝(𝑝 + 1) = 0 are p and -(p+1)

Introduction to Polynomials

If 𝛼, 𝛽 are the zeroes of a polynomial, such that 𝛼 + 𝛽 = 6 and 𝛼𝛽 = 4, then write the polynomial.

x2 - 6x + 4

x2 + 6x + 4

x2 - 3x + 1

x2 - 8x + 9

a

x2 - (𝛼+ 𝛽)x + 𝛼 𝛽

If the zeroes of the quadratic polynomial are 𝛼 and  𝛽 then the quadratic  polynomial is x2 - (𝛼+ 𝛽)x + 𝛼 𝛽 Given  𝛼 + 𝛽 = 6 and 𝛼𝛽 = 4 so substituting values   x2 - (6)x +4  x2 - 6x +4 Hence, The polynomial is  x2 - 6x +4

Relationship between Zeroes and Coefficients of a Polynomial

If one zero of the polynomial 𝑥2 − 4𝑥 + 1 is 2 + √3, write the other zero

2 - √3

2 + √3

3 - √3

3 + √3

a

𝛼 + 𝛽 = -b/a  𝛼𝛽 = c/a

Given polynomial 𝑥2 − 4𝑥 + 1  Let two zeros are  𝛼 and 𝛽  𝛼 = 2 + √3 b = -4 and a = 1 𝛼 + 𝛽 = -b/a 2 + √3 +  𝛽 = 4 𝛽 = 4 - 2 - √3 𝛽 = 2 - √3 Hence, The other zero is  2- √3

Relationship between Zeroes and Coefficients of a Polynomial

For what value of 𝑘, (−4) is a zero of the polynomial 𝑥2 − 𝑥 − (2𝑘 + 2)?
3
9
6
12
b
If (-4) is zero then f(-4) = 0
Given polynomial  𝑥2 − 𝑥 − (2𝑘 + 2) f(x) =  𝑥2 − 𝑥 − (2𝑘 + 2) f(-4) = 0 so  (-4)2 − (-4) − (2𝑘 + 2) = 0 16 + 4 -2k - 2 = 0 18 - 2k = 0 18 = 2k k = 9 Hence, The value of k is 9
Relationship between Zeroes and Coefficients of a Polynomial

Write the polynomial, the product and sum of whose zeroes are − 9/2 and − 3/2 respectively.

2x2 + 9x - 3 = 0

x2 + 18x - 12 = 0

2x2 - 9x - 6 = 0

None of these

a

x2 - (𝛼+ 𝛽)x + 𝛼 𝛽

Given the product zeroes  − 9/2 the sum of zeroes – 3/2 𝛼+ 𝛽 = – 9/2 𝛼 𝛽 = – 3/2 x2 - (𝛼+ 𝛽)x + 𝛼 𝛽 = 0  x2 - (-9/2)x  – 3/2 = 0 multiplying polynomial by 2 2x2 + 9x - 3 = 0

Relationship between Zeroes and Coefficients of a Polynomial

If 1 is a zero of the polynomial 𝑝(𝑥) = 𝑎𝑥2 − 3(𝑎 − 1)𝑥 − 1, then find the value of 𝑎.
1
3
4
2
a
p(1) = 0
Given polynomial 𝑎𝑥2 − 3(𝑎 − 1)𝑥 − 1 p(1) = 0 𝑎𝑥2 − 3(𝑎 − 1)𝑥 − 1 = 0 a(1)2 - 3(a - 1)(1) - 1 = 0 a - 3a + 3 - 1 = 0 -2a + 2 = 0 -2a = -2 a = 1 Hence, The value of a is 1
Relationship between Zeroes and Coefficients of a Polynomial

Is 𝑥 = −3 is a solution of 𝑥2+ 6𝑥 + 9 = 0.

Yes

No

Not enough information

None of the above

a

p(-3) = 0

Given polynomial 𝑥2+ 6𝑥 + 9  if x = -3 is solution then it should satisfy the equation p(-3) = 0 𝑥2+ 6𝑥 + 9 = 0 L.H.S = 𝑥2+ 6𝑥 + 9 => (-3)2 + 6(-3) + 9  => 9 - 18 + 9  => 0 R.H.S = 0 L.H.S = R.H.S Hence, 𝑥 = −3 is a solution of 𝑥2+ 6𝑥 + 9 = 0

Relationship between Zeroes and Coefficients of a Polynomial

If (𝑥 + 𝑎) is a factor of 2𝑥2 + 2𝑎𝑥 + 5𝑥 +10, find 𝑎.
2
3
4
5
a
factor theorem f(-a) = 0
Given polynomial 2𝑥2 + 2𝑎𝑥 + 5𝑥 +10 factor = (x + a) by using factor theorem f(-a) = 0 2(-a)2 + 2a(-a) + 5(-a) + 10 = 0 2a2 - 2a2 -5a + 10 = 0 10 - 5a = 0 10 = 5a a = 2 Hence, The value of a is 2
Factorization of Polynomials

The sum and product of the zeroes of a quadratic polynomial are −1/2 and −3 respectively. What is the quadratic polynomial?

2x2 + x - 6

x2 + x - 6

3x2 + 2x - 16

x2 + 6x - 6

a

x2 - (sum of the roots)x + (product of the roots)

Given sum of the roots = -1/2 product of the roots = -3 x2 - (sum of the roots)x + (product of the roots) => x2 - (-½)x + (-3) => x2 + 1/2x - 3 => 2x2 + x - 6 Hence, The polynomial is  2x2 + x - 6

Relationship between Zeroes and Coefficients of a Polynomial

Polynomials Level 2

a

b

d

Basic knowledge of polynomial

Given, p(x) = ax + b Thyen, zero of p(x), can be written as: Hence, The zero of P(x) is -b/a

Relationship between Zeroes and Coefficients of a Polynomial

1
2
-2
3
b
Given, polynomial is x3 - 3x2 + bx – 6 Also, x – 3 = 0 Now, substituting the value of x in the polynomial, we get Hence, The value of b is 2
Factorization of Polynomials

abc

-abc

3abc

0

c

Simplify the expression

Given, a + b + c = 0 Hence, The value of a3 + b3 + c3 is 3abc

Introduction to Polynomials

a

Convert the expression in two factors

Given, x2 + 5x + 6 Hence, The possible dimension is (x + 3) (x + 2)

Application of Polynomials in solving Real-Life Problem

0

1

a

-a

c

α and 1/α = c/a

Given, α and 1/α are the zeroes of the polynomial ax2 + bx + c Hence, The value of c is a

Relationship between Zeroes and Coefficients of a Polynomial

-16
8
7
4
a
Basic knowledge of polynomial
Hence, The value of k is -16
Relationship between Zeroes and Coefficients of a Polynomial

7x

14x

c

Simplify the expression

Given, (1 + 7x)2 + (49x2 - 1) Hence, 14x is one of the factors of (1 + 7x)2 + (49x2 - 1)

Introduction to Polynomials

Dividend is equal to

divisor × quotient + remainder

divisor × quotient 10

divisor × quotient - remainder

divisor × quotient × remainder

a

Basic knowledge of math

Dividend = divisor × quotient + remainder Hence, The answer is divisor x quotient + remainder

Introduction to Polynomials

The maximum number of zeros that a polynomial of degree 4 can have, is
1
2
3
4
d
The degree of polynomial indicates the number of zeros.  Hence, The answer is 4
Since we know that the degree of polynomial indicates the number of zeros.  So maximum no. of zeros a polynomial of degree 4 can have is 4.
Degree of a Polynomial

1
2
3
4
c
Zeroes are the x-intercepts of the graph
Since we know that zeroes are the x-intercepts of the graph.  Also, we can clearly see that there are 3 x-intercepts in this graph.  Hence, The zeroes of the polynomial f(x) is 3.
Relationship between Zeroes and Coefficients of a Polynomial

(1, 2)
(-1, 2)
(1, -2)
(-1, -2)
d
Factorize the expression
We have,  Hence, The zeroes are -1 and -2
Relationship between Zeroes and Coefficients of a Polynomial

(2, 5)
(-2, 5)
(2, -5)
(-2, -5)
d
Factorize the expression
We have, Hence, The zeroes are -2 and -5
Relationship between Zeroes and Coefficients of a Polynomial

2
3
4
5
d
Simplify the expression
Given, polynomial is (x + 1)(x2 – x4 + 1) Hence, The degree of the polynomial is 5
Degree of a Polynomia

Value of p(x)

Zero of p(x)

Constant term of p(x)

None of above

b

As 0 is the remainder that means (x - k) completely divides the polynomial p(x)

Given, p(x) is a polynomial of at least degree 1, highest power of variable x is at least one.   As 0 is the remainder that means (x - k) completely divides the polynomial and we can say k is the zero of the polynomials.  Hence, k is the zero of p(x)

Introduction to Polynomials

Both positive

Both negative

One positive one negative

Both equal

c

Factorize the expression

Given, polynomial is 3x2 – 48 So, we have 2 zeroes one of which is positive and one is negative. Hence, The zeroes of the polynomaisk are 4 and -4

Relationship between Zeroes and Coefficients of a Polynomial

Which of the following is the cubic polynomial?

x5 + y3 –x3 + x4

x3 + y3 –x2

2x3 + 8y3 -9x5

-7x7 + x3

b

A cubic polynomial is a polynomial of degree 3.

Since we know that, a cubic polynomial is a polynomial of degree 3 So, we can see that the polynomial x3 + y3 –x2 is the cubic polynomial as in this polynomial, the highest degree is 3. Hence, x3 + y3 –x2  is the correct answer

Types of Polynomials (Constant, Linear, Quadratic, Cubic, etc.)

p(x) = 63 – 9x is___ a polynomial.

Linear

Cubic

Quadratic

Linear

d

A polynomial having highest degree one is known as linear polynomial.

Given, p(x) = 63 – 9x We can see that the above polynomial has degree 1.  Hence, The polynomial p(x) = 63 – 9x is a linear polynomial

Types of Polynomials (Constant, Linear, Quadratic, Cubic, etc.)

Which of the following is a polynomial with only one zero?

P(x) = 2x2 – 3x + 4

P(x) = x2 – 2x + 1

P(x) = 2x – 3

P(x) = 5

c

Linear polynomial has only one zero

We know that, linear polynomial has only one zero. Out of given options, p(x) = 2x + 3 is only linear polynomial. Hence, The answer is P(x) = 2x – 3

Relationship between Zeroes and Coefficients of a Polynomial

The number of zeroes in the given graph y = p(x) is
2
0
6
8
b
The graph does not intersects the x – axis
Given, the graph is The number of zeroes in the given graph is zero as the graph does not intersects the x – axis.  Hence, The answer is 0
Relationship between Zeroes and Coefficients of a Polynomial

The linear polynomial p(x) = 7x – 3 has the zero for

2/3

3/9

3/7

-3/7

c

Substitute all values one by one in the given equation

Given, p(x) = 7x – 3 p(x) is 0 for x = 3/7 When we put 3/7 in the equation, we get p(3/7) = 7(3/7)– 3 = 3- 3 = 0 Hence, The linear polynomial p(x) = 7x – 3 has the zero for 3/7

Relationship between Zeroes and Coefficients of a Polynomial

Write the correct alternative answer for the following question: Which is the degree of the polynomial 2x2 + 5x3 + 7?
2
5
3
7
c
The highest power of the polynomial term with a non – zero coefficient, is called the degree of the polynomial
The given polynomial is 2x2 + 5x3 + 7, We know, in a polynomial in one variable, the highest power of the polynomial term with a non – zero coefficient, is called the degree of the polynomial. Hence, the highest power of the polynomial variable x is 3. Since the degree of the given polynomial is 3. Hence, The degree of the polynomial 2x2 + 5x3 + 7 is 3
Degree of a Polynomial

Write the correct alternative answer for the following question: Which of the following is a linear polynomial?

x2 + 5

x4 + 5

x3 + 5

x + 5

d

The general form of a linear polynomial is ax + b where, a≠0

The general form of a linear polynomial is ax + b where, a≠0 The polynomial with degree 1 is called linear polynomial. So, here x + 5 is the linear polynomial. Hence, The answer is x + 5

Types of Polynomials (Constant, Linear, Quadratic, Cubic, etc.)

The degree of the polynomial X4 + X3 is:
2
3
4
5
c
The highest power of the variable in a polynomial in one variable is called the degree of the polynomial.
Since we know that the highest power of the variable in a polynomial in one variable is called the degree of the polynomial. Here, in the given polynomial X4 + X3, The highest power of the variable x is 4. Hence, The degree of the given polynomial is 4.
Degree of a Polynomial

If the degree of the dividend is 5 and the degree of the divisor is 3, then the degree of the quotient will be:
-2
1
0
2
d
Degree of quotient is degree of the divided minus degree of the divisor
Recall, f(x) = g(x) q(x) + r(x) given that the degree of the divided is 5 and the degree of the divisor is 3.  Degree of quotient is 5 – 3 = 2. Hence, The degree of the quotient will be 2
Division Algorithm for Polynomials

The polynomial 9x2 + 6x + 4 has

Many real zeroes

Two real zeroes

No real zeroes

One real zeroes

c

The given polynomial cannot be factorized

The polynomial 9x2 + 6x + 4 has no real zeroes because it cannot be factorize. Hence, The answer is no real zeroes

Relationship between Zeroes and Coefficients of a Polynomial

The sum of two zeroes of the polynomial f(x) = 2x2 + (p + 3) x + 5 is zero, and then the value of ‘p’ is
3
-3
-4
4
b
Sum of zeroes = -b/a
Given, polynomial f(x) = 2x2 + (p + 3) x + 5 Let zeroes of the given polynomial be α and β According to the question, α + β = 0 -b / a = 0 -(p+3) / 2 = 0 -(p+3) = 0 (by cross multiplication) P = -3 Hence, The value of p is -3
Relationship between Zeroes and Coefficients of a Polynomial

Which of the following is a cubic polynomial?

x3 + 3x2 – 4x + 3

x2 +  4x – 7

3x2 + 4

3(x + x2 + 1)

a

A cubic polynomial is a polynomial of degree 3

A cubic polynomial is a polynomial of degree 3, or in simpler words highest power of x is 3. It is of the form f (x) = ax3 + bx2 + cx + d where a, b, c, d belongs to real numbers. We can see that in x3 + 3x2 – 4x + 3, x has the highest power 3 and a = 1, b= 3, c = -4 and d = 3. So, it is a cubic polynomial. Rest is the quadratic polynomials. Hence, The answer is x3 + 3x2 – 4x + 3

Types of Polynomials (Constant, Linear, Quadratic, Cubic, etc.)

Degree of which of the following polynomial is zero?

x

15

y

x + x2

b

The degree of a polynomial is the highest degree of its monomials ( individual terms) with non – zero coefficients

The degree of a polynomial is the highest degree of its monomials (individual terms) with non – zero coefficients. For, non – zero constants, degree is zero. Here, 15 are the only non – zero constant and thus has zero degree. Hence, The answer is 15

Degree of a Polynomial

If sum of the roots is 2 and product is 5, then the quadratic equation is?

x2 + 2x + 5 = 0

x2 - 2x - 5 = 0

x2 + 2x - 5 = 0

x2 - 2x + 5 = 0

d

The quadratic equation is given by x2 – (sum of the roots) x + (product of the roots)

Given, the sum of the roots is 2 and the product of the roots is 5. Since we know that, the quadratic equation is given by x2 – (sum of the roots) x + (product of the roots) = 0. Hence, x2 - 2x + 5 = 0 is the quadratic equation.

Relationship between Zeroes and Coefficients of a Polynomial

Find the degree of the given algebraic expression: 3x – 15?
1
3
2
-15
a
x0 = 1
Given, 3x – 15 has two terms.  The first term is 3x and the second term is – 15. The exponent of the first term is 1 because 3x = 3x1 The exponent of the second term is 0 because 15 = 15x0 Since the highest exponent is 1.  Hence, The degree of the algebraic expression 3x – 15 is 1.
Degree of a Polynomial

Degree of the polynomial 13 + 11x + 12x3 + 3x2 is?
2
1
4
3
d
The degree of an individual term of a polynomial is the exponent of its variable
The degree of an individual term of a polynomial is the exponent of its variable. 13 + 11x + 12x3 + 3x2  = 12x3 + 3x2 + 11x + 13 The highest exponent of x is 3, as the degree is 3. Hence, The degree of the polynomial 13 + 11x + 12x3 + 3x2 is 3
Degree of a Polynomial

The constant polynomial whose coefficients are all equal to 0 is called _______ polynomial.

Zero

Linear

Quadratic

Cubic

a

Zero polynomial can be written as p(x) = 0

Consider the polynomial, p(x) = ax2 + bx + c, if a = b = c = 0 then the expression becomes zero polynomial. Therefore, zero polynomial can be written as p(x) = 0. Hence, The constant polynomial whose coefficients are all equal to 0 is called a zero polynomial.

Relationship between Zeroes and Coefficients of a Polynomial

The degree of the polynomial 5x7 – 6x5 + 7x – 6 is
7
5
6
4
a
The degree of a polynomial is the highest exponent of variable
Since we know that the degree of a polynomial is the highest exponent of variable. Here highest exponent is 7. Hence, The degree of the polynomial 5x7 – 6x5 + 7x – 6 is
Degree of a Polynomial

Which of the following is Quadratic polynomial?

X + 2

X2 + 2

X3 + 2

2x + 2

b

A Quadratic polynomial is a polynomial of degree 2

We know that, a quadratic polynomial is a polynomial of degree 2. Hence, x2 + 2 is the only quadratic polynomial among all the four given polynomials.

Types of Polynomials (Constant, Linear, Quadratic, Cubic, etc.)

The polynomial 4x2 + 2x – 2 is a

Linear polynomial

Cubic polynomial

Constant polynomial

Quadratic polynomial

d

Highest power in the given polynomial is 2

Given, 4x2 + 2x – 2 We can see that in the given polynomial, the highest power is 2.  Hence, The polynomial 4x2 + 2x – 2 is a quadratic polynomial

Types of Polynomials (Constant, Linear, Quadratic, Cubic, etc.)

Which type of polynomial y3 – y?

Linear polynomial

Quadratic polynomial

Cubic polynomial

None of these

c

The given polynomial has the highest power of 3

Given, y3 – y is a polynomial which has the highest power of x is 3. Hence, The given polynomial is a cubic polynomial

Types of Polynomials (Constant, Linear, Quadratic, Cubic, etc.)

A polynomial of degree _______________ is called a linear polynomial?
0
1
2
3
B
A polynomial of degree 1 is called a linear polynomial
A polynomial of degree 1 is called a linear polynomial. For example: 4x + 3, 65y are linear polynomial.
Degree of a Polynomial

Which type of polynomial is 1 + x + x2?

Linear polynomial

Cubic polynomial

Quadratic polynomial

None of these

c

The degree of the equation is 2

Given,  1 + x + x2 The degree of the equation is 2.  Hence, It is a Quadratic polynomial.

Types of Polynomials (Constant, Linear, Quadratic, Cubic, etc.)

Classify the following as a constant, linear, quadratic, and cubic polynomial: √2x + 1

Cubic

Quadratic

Linear

None of these

c

In the expression √2x + 1  the degree of x is 1

Given, √2x + 1 Here, the degree of x is 1.  Hence, It is a linear polynomial.

Types of Polynomials (Constant, Linear, Quadratic, Cubic, etc.)

The degree of the polynomial p(x) = x3 -9x + 3x5 is
3
0
5
2
c
The degree of a polynomial is the highest power of any of its variable
Given, polynomial p(x) = x3 -9x + 3x5 We know that, the degree of a polynomial is the highest power of any of its variable. Here, in p(x) = x3 -9x + 3x5, the highest power of variable x is 5. So, the degree is 5. Hence, The degree of p(x) = x3 -9x + 3x5 is 5.
Degree of a Polynomial

Which type of polynomial is 45y2?

Linear polynomial

Bi – quadratic polynomial

Cubic polynomial

Quadratic polynomial

d

The highest power in the given polynomial is 2

Given, 45y2  The given polynomial is a quadratic polynomial as the highest power of y is 2. Hence, The answer is quadratic polynomial

Introduction to Polynomials

A polynomial of degree_______________ is called a cubic polynomial?
0
1
2
3
d
The standard form of cubic polynomial is f (x) = a3x3 + a2x2 + a1x + a0
A polynomial of degree 3 is called a cubic polynomial.  A cubic polynomial has the form f (x) = a3x3 + a2x2 + a1x + a0. An equation involving a cubic polynomial is called a cubic equation. Hence, A polynomial of degree 3 is called a cubic polynomial
Introduction to Polynomials

The largest power of ‘x’ in p (x) is the _____________ of the polynomial.

Root

Zero

Degree

Solution

c

A degree in a polynomial function is the greatest exponent of that equation

A degree in a polynomial function is the greatest exponent of that equation.  Therefore, the degree of the constant is zero. Hence, The largest power of ‘x’ in p (x) is the degree of the polynomial

Introduction to Polynomials

The value of quadratic polynomial f (x) = 2x2 – 3x -2 at x = -2 is……
12
-12
15
16
a
Substitute the value of x = -2 in the given polynomial f(x)
Given, f (x) = 2x2 – 3x -2  Now by substituting x = 2 in 2x2 – 3x -2, we get = 2(-2)2 – 3 (-2) - 2  = 8 + 6 – 2  = 12 Hence, The value of quadratic polynomial f (x) = 2x2 – 3x -2 at x = -2 is 12
Relationship between Zeroes and Coefficients of a Polynomial

The number of polynomials having zeroes -2 and 5 is

1

2

3

More than 3

d

We can write the polynomial having -2 and 5 as the zeroes in the form of k (x + 2) (x – 5), where k is a constant.

The polynomial having -2 and 5 as the zeroes can be written in the form k (x + 2) (x – 5), where k is a constant. Thus, number of polynomials with roots -2 and 5 are infinitely many, since k can take infinitely many values. Hence, The number of polynomials having zeroes -2 and 5 is more than 3.

Relationship between Zeroes and Coefficients of a Polynomial

If ‘1’ is a zero of the polynomial p (a) = x2a2 – 2xa + 2x – 4, then x = _________?

+2, -2

4

-2, 0

-3

a

Solve for p(1) = 0

Given, p (a) = x2a2 – 2xa + 2x – 4 Since 1 is a zero, so p (1) = 0 p (1) = x212 – 2x1 + 2 x – 4 = 0 x2 – 4 = 0 x = +2, -2 Hence, The x is +2 and -2

Introduction to Polynomials

The zeroes of x2 -2x – 8 are
(2, -4)
(4, -2)
(-2, -2)
(-4, -4)
b
Solve by factorization
Given, x2 -2x – 8  = x2 – 4x + 2x – 8 = x(x – 4) + 2 (x – 4) = (x – 4) (x + 2)  x = 4, -2. Hence, The zeroes of x2 -2x – 8 are 4 and -2
Relationship between Zeroes and Coefficients of a Polynomial

What is the quadratic polynomial whose sum and the product of zeroes is √2, ½ respectively?

3x2 – 3 √2x + 1

3x2 + 3 √2x + 1

3x2 + 3 √2x – 1

None of the above

a

If α and  β are zeroes of any quadratic polynomial, then the polynomial is, x2 = (α + β)x + αβ

Given, sum of zeroes = α + β = √2 And product of zeroes = α × β = 1/3 If α and  β are zeroes of any quadratic polynomial, then the polynomial is, x2 = (α + β)x + αβ =x2 – (√2)x + (1/3) = 3x2 – 3 √2x + 1  Hence, The polynomial is 3x2 – 3 √2x + 1

Relationship between Zeroes and Coefficients of a Polynomial

A polynomial of degree n has

Only one zeroes

At least n zeroes

More than n zeroes

At most n zeroes

d

Maximum number of zeroes of a polynomial = degree of the polynomial.

We know that, maximum number of zeroes of a polynomial is equal to the degree of the polynomial. Hence, A polynomial of degree n has at most n zeroes.

Introduction to Polynomials

Zeroes of p(x) = x2 – 27 are

+ 9√3, - 9√3

+ 3√3, - 3√3

+ 7√3, -7√3

None of the above

b

Solve for p(x)= 0

Given, p(x) = x2 – 27 x2 – 27 = 0 x2 = 27 x= √27 x = + - 3√3 Hence, Zeroes of p(x) = x2 - 27 are + 3√3 and - 3√3

Relationship between Zeroes and Coefficients of a Polynomial

Given that two of the zeroes of the cubic polynomial ax3 + bx2 + cx + d are 0, the third zero is

-b/a

b/a

c/a

d/a

a

Sum of the zeroes = -b/a

Given, ax3 + bx2 + cx + d and two of its zeroes are 0  Let α be the third zero. Sum of the zeroes = α + 0 + 0 = -b/a α = -b/a. Hence, The third zero is -b/a

Relationship between Zeroes and Coefficients of a Polynomial

If the graph of a polynomial intersects the x – axis at three points, then it contains_______ zeroes.

Three

Two

Four

More than three

a

The point where the polynomial cuts the x-axis is the zero of the polynomial.

If the graph of a polynomial intersects the x – axis at three points, then it contains three zeroes. Hence, The answer is three

Relationship between Zeroes and Coefficients of a Polynomial

Sum of the squares of the zeroes of the polynomial p(x) = x2 + 7x – k is 25, find k.
12
49
-24
-12
d
Solve for 49 + 2k = 25
Given, p(x) = x2 + 7x – k Sum of zeroes of the polynomial is 25   49 + 2k = 25 2k = 25 – 49 k = -24/2 k = -12 Hence, The value of k is -12
Introduction to Polynomials

A real number ‘k’ is said to be a zero of a polynomial p(x), if p(k) =
2
1
3
0
d
The zero of the polynomial is defined as any real value for which the value x of the polynomial becomes zero
The zero of the polynomial is defined as any real value for which the value x of the polynomial becomes zero. A real number k is a zero of a polyn0omial p(x), if p (k) = 0 For example; p (x) = x2 – 3x – 4 Then p(-1) = (-1)2 – (3 × -1) – 4 = 0 And p(4) = (4)2 – (3 × 4) – 4 = 0 For a polynomial p(x), real number k is said to be zero of polynomial p (x), if p (k) = 0. Hence, The answer is 0
Relationship between Zeroes and Coefficients of a Polynomial

If ‘α’, ‘β’ and ‘ϒ’ are the zeroes of a cubic polynomial ax3 + bx2 + cx + d, then αβ + βϒ + ϒα is

b/a

-b/a

-c/a

c/a

d

Product of zeroes of a cubic polynomial ax3 + bx2 + cx + d = (coefficient of x) / coefficient of x3

Given, ‘α’, ‘β’ and ‘ϒ’ are the zeroes of a cubic polynomial ax3 + bx2 + cx + d. Therefore, sum of the product of zeroes of a cubic polynomial ax3 + bx2 + cx + d = (coefficient of x) / coefficient of x3 Then, αβ + βϒ + ϒα = c/a Hence, The answer is c/a

Relationship between Zeroes and Coefficients of a Polynomial

A polynomial whose sum and product of zeroes are -4 and 3 is

x2 – 4x + 3

x2 – 4x – 3

x2 + 4x + 3

None of these

c

Required polynomial = x2 – ( sum the zeroes ) x + (product of zeroes)

Given, sum and product of zeroes are -4 and 3 respectively Required polynomial = x2 – ( sum the zeroes ) x + (product of zeroes) x2 – (-4)x + 3 = x2 + 4x + 3 Hemce, The polynomial is  x2 + 4x + 3

Relationship between Zeroes and Coefficients of a Polynomial

The sum of two zeroes of the polynomial f (x) = 2x2 + (p+3) x + 5 is zero, then the value of ‘p’ is
-4
-3
3
4
b
Sum of two zeroes of the polynomial = -b/a
Given, f (x) = 2x2 + (p+3) x + 5 Let zeroes of the given polynomial be α and β According to the question, α+ β = 0 -b/a = 0 -(p + 3) / 2 = 0 -(p + 3) = 0 ( by cross multiplication) p = -3 Hence, The value of p is -3
Relationship between Zeroes and Coefficients of a Polynomial

If two zeroes of the polynomial 𝑥3 − 4𝑥2 − 3𝑥 +12 are √3 and −√3, then find its third zero.
1
2
3
4
d
++ = -b/a
Given polynomial 𝑥3 − 4𝑥2 − 3𝑥 +12 a = 1, b = -4, c = -3, d = 12 =√3  , =−√3 ++ = -b/a  √3 − √3  + = 4/1 = 4 Hence, The third zero is 4
Relationship between Zeroes and Coefficients of a Polynomial

If the polynomial 6𝑥4 + 8𝑥3 + 17𝑥 2 + 21𝑥 + 7 is divided by another polynomial 3𝑥2 + 4𝑥 + 1, the remainder comes out to be (𝑎𝑥 + 𝑏), find 𝑎 and 𝑏.

(1,2)

(2,2)

(1,1)

None of these

a

divide polynomial

6𝑥4 + 8𝑥3 + 17𝑥 2 + 21𝑥 + 7 3𝑥2 + 4𝑥 + 1 after dividing remainder is x + 2 comparing with ax + b  a = 1 and b = 2 Hence, The a and b are 1 and 2 respectively

Factorization of Polynomials

Polynomials Level 3

a

Factorize the expression

We have, Hence, The zeroes of 6x2 - 7x - 3 are 3/2 and -1/3

Factorization of Polynomials

Both positive

Both negative

One positive and one negative

Both equal

c

Factories the expression

We have,  We can clearly see that one of the zeroes is positive and one is negative. Hence, The zeroes of the quadratic polynomial x2 - 2x - 8 are one positive and one negative

Factorization of Polynomials

Both positive

Both negative

One positive and one negative

Both equal

a

Factorize the expression

We have,  So, both the zeroes are positive. Hence, The zeroes of the quadratic polynomial x2 - 15x + 50 are both positive

Factorization of Polynomials

Both positive

Both negative

One positive and one negative

Both equal

b

Sum of both roots = -b/a and product of roots = c/a

We have, x2 + 1750x + 175000 On comparing the given polynomial with ax2 + bx + c, we get Here, a = 1, b = 1750 and c = 175000 Since, sum of both roots = -b/a  = -1750/1 = -1750, which is negative Similarly, product of roots = c/a  = 175000/1  = 175000, which is postive number  Hence, The zeroes of the quadratic polynomial x2 + 1750x + 175000 are both negative

Application of Polynomials in solving Real-Life Problems

Both positive and unequal

Both negative

One positive and one negative

Both equal and positive

d

Factorize the expression

We have,  So, both the zeroes are positive. Hence, The zeroes of the quadratic polynomial x2 - 18x + 81 are both equal and positive

Factorization of Polynomials

A quadratic polynomial, whose zeroes are -4 and -5, is

a

Basic knowledge of polynomial

We have the roots of polynomial which are -4 and -5  So, Hence, A quadratic polynomial, whose zeroes are -4 and -5, is x2 + 9x + 20

Relationship between Zeroes and Coefficients of a Polynomial

-2, 5
2, 5
2, -5
-2, -5
d
Factorize the expression
We have, Hence, -2 and -5 are the zeroes of the polynomial x2 + 7x + 10
Factorization of Polynomials

If the zeroes of the quadratic equation are 11 and 2, what is expression for quadratic

b

Basic knowledge of polynomial

Given, zeroes of the quadratic equation are 11 and 2 Hence, The expression is x2 - 13x + 22

Relationship between Zeroes and Coefficients of a Polynomial

3, 4

-3, 4

b

Factorize the expression

We have,  Hence, The zeroes of the polynomial 4x2 - 12x + 9 are 3/2 and 3/2

Factorization of Polynomials

The number of polynomials having zeroes as 4 and 7 is

2

3

4

More than 4

d

Sum of both roots = -b/a and product of roots = c/a

Hence, The number of polynomials having zeroes as 4 and 7 is more than 4

Relationship between Zeroes and Coefficients of a Polynomial

Zeroes of a polynomial can be expressed graphically. Number of zeroes of polynomial is equal to number of points where the graph of polynomial is:

Intersects x-axis

Intersects y-axis

Intersects x-axis or y-axis

None of above

a

Zeros are the point where the curve intersects the x-axis

Since we know that zeros are the point where the curve intersects the x-axis Therefore, number of zeroes of polynomial is equal to number of points where the graph of polynomial intersects x-axis. Hence, The answer is intersects x-axis

Introduction to Polynomials

Both positive

Both negative

One positive and one negative

Both equal

b

Sum of both roots = -b/a and product of roots = c/a

Given, x2 + 99x + 127 Using the quadratic formula, x = [-b + (√b2 – 4ac)]/ 2a; x = [-b – (√b2 – 4ac)]/ 2a Here, a = 1, b = 99 and c = 127 x = -1.3 and x = -99.7 So, both the roots are negative Hence, The zeroes of x2 + 99x + 127 are both negative

Relationship between Zeroes and Coefficients of a Polynomial

The quadratic polynomial whose sum of zeroes is 3 and product of zeroes is 2 is

d

Sum of zeroes = -b/ a and product of zeroes = c/a

Hence, The quadratic polynomial whose sum of zeroes is 3 and product of zeroes is 2 is x2 - 3x + 2

Relationship between Zeroes and Coefficients of a Polynomial

c and a have same sign

c and b have same sign

c and a have opposite sign

c and b have opposite sign

a

For equal roots, b2 – 4ac = 0

Given, roots of the polynomial x2 + bx + c, have equal roots Hence, The answer is c and a have same sign

Relationship between Zeroes and Coefficients of a Polynomial

Graph of a quadratic polynomial is a

Straight line

Parabola

Circle

Ellipse

b

The graph of a quadratic function is a U-shaped curve

The graph of a quadratic function is a U-shaped curve called a parabola. Hence, The answer is parabola

Introduction to Polynomials

10
-10
5
-5
b
Given, x2 + 3x + k And one of the zeroes of the polynomial is 2 So,  Hence, The value of k is -10
Relationship between Zeroes and Coefficients of a Polynomial

5/3

a

Given, (p - 1)x2 + px + 1 and one of its zeroes is -3 Now,  Hence, The value of p is 4/3

Relationship between Zeroes and Coefficients of a Polynomial

A polynomial of degree n has

Only 1 zero

At most n zeroes

More than n zeroes

Exactly n zeroes

b

Basic knowledge of polynomial

A polynomial of degree n can have at most n number of zeroes. Hence, The answer is at most n zeroes

Introduction to Polynomials

2
-2
4
-4
b
Given, x2 + x + k and 1 is one of its zeroes So substituting the value of x = 1, we get Hence, The value of k is -2
Relationship between Zeroes and Coefficients of a Polynomial

Which one of the following is an example of cubic polynomial?

x5 + y3 –x3 + x4

x5 + y3 –x3

x3 + x4

x3 + 1 =0

d

A cubic polynomial is a polynomial of degree 3

Since we know that, a cubic polynomial is a polynomial of degree 3. The highest degree should be 3 in order to be a cubic polynomial. So, x3 + 1 = 0 is an example of a cubic polynomial  Hence, The answer is x3 + 1 = 0

Types of Polynomials (Constant, Linear, Quadratic, Cubic, etc.)

If two roots of the equation x2 - 3x + 2 = 0 are same, then the roots will be
1, 1, -2
2, 8, 4
-1, 4, 8
-2, -2, -2
a
x = 1 is a solution of the equation (by observation)
Given, x2 - 3x + 2 = 0, which have two common roots x = 1 is a solution of the equation (by observation) Therefore, x – 1 is a factor of equation So, on dividing x2 - 3x + 2 by x – 1, we get x2 + x - 2  So, x2 + x – 2 = 0 x2 – 2x- x – 2 = 0 x(x + 2) – 1(x + 2) = 0 x = 1, x = -2 Hence, The roots will be 1, 1, -2
Relationship between Zeroes and Coefficients of a Polynomial

The quadratic polynomials f(x) = ax2 + bx + c has real zeroes α and β. If a, b, c are real and of the same sign then

α > 0, β < 0

β > 0, α > 0

α < 0, β > 0

β < 0, α < 0

d

a, b, c have same sign so, α  + β < 0

Given, f(x) = ax2 + bx + c Let f(x) = 0, then α  + β = -b/a α β = c/a since, a, b, c have same sign  α  + β < 0 α β > 0 Since, α, β are real roots Therefore, α, β both are negative So, α < 0 and  β < 0 Hence, The answer is β < 0, α < 0

Introduction to Polynomials

If α, β are zeroes of p(x) = 2x2 + 5x + 5, find the value of α-1 + β-1.

½

1

-2

-1

d

α + β = -5/2 as α, β are the roots of the given equation

Given, p(x) = 2x2 + 5x + 5 α, β are the roots of the given equation α + β = -5/2 1/ α + 1/β = (β + α)/ αβ (α+ β)/ αβ = (-5/2)/ (5/2) = -1 1/ α + 1/β = -1 α-1 + β-1 = -1  Hence, The value of α-1 + β-1 is -1

Relationship between Zeroes and Coefficients of a Polynomial

The graph of a polynomial is shown in given figure, then the number of its zeroes is
1
2
3
4
c
The number of zeroes are the value at which for any value of x, y is zero
Since we know that, the number of zeroes is the value at which for any value of x, y is zero. Here at 3 points y becomes 0, so number of zeroes are 3 Hence, The answer is 3
Introduction to Polynomials

The graph of the polynomial p(x) intersects the x – axis three times in distinct points, then which of the following could be an expression for p (x):

3x + 3

3x2 + 3x -3

X2 -9

4 -4x – x2 + x3

d

The graph of the polynomial intersects the x – axis at three distinct points so, it will have 3 distinct roots.

Since, the graph of the polynomial p(x) intersects the x – axis at three distinct points, it will have 3 distinct roots. it will be a cubic polynomial. Hence, Amongst the given alternatives the polynomial p (x) can be 4 -4x – x2 + x3.

Relationship between Zeroes and Coefficients of a Polynomial

If a real number ‘a’ is a zero of a polynomial, then __________ is a factor of f(x)?

x + a

x × a

x - + a

x – a

d

Basic knowledge of polynomial

If a real number a is a zero of a polynomial, then(x – a) is a factor of that polynomial. (By factor theorem). Hence, The answer is x - a

Introduction to Polynomials

The graph of the polynomial f(x) = 2x – 5 intersects the x –axis at

(5/2, 5/2)

(5/2, -5/2)

(5/2, 0)

(-5/2, 0)

c

When the graph is on x- axis then the y point (ordinate) is 0

Graph f(x) = 2x – 5 is on x – axis (when the graph is on x- axis then the y point (ordinate) is 0) Put f(x) = 2x – 5 = 0 2x = 5 x = 5/2  So, the coordinates of graph is (5/2, 0) Hence, The graph of the polynomial f(x) = 2x - 5 intersects x-axis at (5/2, 0)

Relationship between Zeroes and Coefficients of a Polynomial

The number of zeroes that the polynomial f(x) = (x – 2)2 + 4 can have is
0
1
2
3
b
Simplify the expression and solve
Given, f(x) = (x – 2)2 + 4  = x2 – 4x + 8, here the largest exponent of variable is 2 Hence, The number of zeroes of the given polynomial is 2.
Relationship between Zeroes and Coefficients of a Polynomial

When we divide x3 + 5x + 7 by x4 -7x2 – 6 then quotient and remainder are

0, x3 + 5x + 7

X, 2x + 3

1, x4 – 7x2 – 6

X2, 4x – 9

a

Degree of the divisor is more than the degree of the dividend

Degree of the divisor is more than the degree of the dividend ,  quotient is zero and the remainder is x3 + 5x + 7 Hence, The answer is 0, x3 + 5x + 7

Division Algorithm for Polynomials

If one zero of the polynomial p(x) = (k + 4) x2 + 13x + 3k is reciprocal of the other, then the value of ‘k’ is?
4
3
5
2
d
Product of roots, αβ = c/a
Let one zero of the given polynomial be α and the other zero is reciprocal be 1/ α. Since (products of roots) αβ = c/a  α × 1/ α  = 3k / k + 4 1 = 3k / k + 4 K + 4 = 3k (by cross multiplication)  4 = 3k – k  4 = 2k  k = 4/2  k = 2 Hence, The value of k is 2
Relationship between Zeroes and Coefficients of a Polynomial

The degree of the polynomial 2 – y2 – y3 + 2y7 is
2
7
0
3
b
The degree of a polynomial is the highest degree of its monomials (individual terms) with non – zero coefficients
Since we know that, the degree of a polynomial is the highest degree of its monomials (individual terms) with non – zero coefficients. The degree of a term is the sum of the exponents of the variable that appear in it, and thus is a non – negative integer. Hence, for 2 – y2 – y3 + 2y7, the degree is 7.
Degree of a Polynomial

Which of the following is a quadratic polynomial in one variable?

√2x3 + 5

2x2 + 2x -2

x2

2x2 + y2

c

A quadratic polynomial is a polynomial of degree 2

Since we know that, a quadratic polynomial is a polynomial of degree 2. Here, x2 is a polynomial of degree 2.  Hence, The answer is x2

Types of Polynomials (Constant, Linear, Quadratic, Cubic, etc.)

Find x, if 16.7 + 12.38 – x = 10.09?
17.89
18.99
16.98
20.09
b
Solve by method of transposition
Given, 16. 7 + 12. 38 – x = 10.09  Shifting x to the RHS, we get  16. 7 + 12. 38 – 10.09 = x x = 18.99 Hence, The value of x is 18.99
Introduction to Polynomials

The graph of linear polynomial p(x) = 3x – 6 intersects the x – axis at ___________
(0, 6)
(3, -6)
(0, 2)
(2, 0)
d
Solve for 3x – 6 = 0
For intersecting at x – axis, p(x) = 0 3x – 6 = 0 x = 6/3 x = 2 Hence, The graph of linear polynomial p(x) = 3x – 6 intersects x-axis at (2, 0)
Introduction to Polynomials

If one of the zeroes of the quadratic polynomial (k – 1)x2 + kx + 1 is -3, then the value of k is

4/3

-4/3

2/3

-2/3

a

-3 must satisfy the equation (k -1)x2 + kx + 1 = 0

Given, -3 is the zero of the polynomial (k-1)x2 + kx + 1 So, -3 must satisfy the equation (k -1)x2 + kx + 1 = 0 (k – 1) (-3)2 + k(-3) + 1 = 0 9 (k – 1) – 3k + 1 = 0 6k = 8 k = 4/3 Hence, The value of k is 4/3

Relationship between Zeroes and Coefficients of a Polynomial

The degree of the polynomial 4x2 – 7x3 + 6x + 1 is
2
3
1
0
b
Degree of a polynomial is the highest power of x in its expression
A polynomial is a function of the form f (x) = an xn + an – 1x n – 1 + … + a2x2 + a1x + a0 where an, an -1… a2, a1, a0 are constants and n is a natural number. Let p(x) = 4x2 – 7x3 + 6x + 1 the degree of a polynomial is the highest power of x in its expression. Highest power of x in p(x) = 3 Thus, the degree = 3 Therefore, p(x) = 4x2 – 7x3 + 6x + 1 is a cubic polynomial. Hence, The degree of the polynomial 4x2 – 7x3 + 6x + 1 is 3
Degree of a Polynomial

The degree of the polynomial p (x) = x7 -5x3 – 3x2 + 2x is _____________?
2
3
7
1
c
Given,  polynomial is p(x) = x7 -5x3 – 3x2 + 2x Now the power of the highest degree term is 7  Hence, The degree of the polynomial is 7.
Degree of a Polynomial

Which type of polynomial are 2 – x2 + x3?

Binomial

Cubic polynomial

Quadratic polynomial

None of these

b

The highest power of x in the given polynomial is 3

Given, 2 – x2 + x3  We can see that the highest power is 3 So, it is a cubic polynomial as the highest power of x is 3. Hence, The polynomial is cubic polynomial

Types of Polynomials (Constant, Linear, Quadratic, Cubic, etc.)

A polynomial of degree ‘m’ has

‘m’ zeroes

One zero

At least ‘m’ zeroes

At most ‘m’ zeroes

d

Degree of a polynomial is equal to the zeroes of that polynomial only.

A polynomial of degree ‘m’ has at most ‘m’ zeroes because degree of a polynomial is equal to the zeroes of that polynomial only. Hence, The answer is at most m zeroes

Introduction to Polynomials

If one of the zeros of f (x) = x3 + 13x2 + 32x + 20 is -2 then all its zeros are
2, 13, 11
4, -2, -10
-2, 5, 10
-10, -1, -2
d
Divide f (x) = x3 + 13x2 + 32x + 20 by (x + 2)
Given, f (x) = x3 + 13x2 + 32x + 20 One of its zeroes is -2  By dividing f (x) = x3 + 13x2 + 32x + 20 by (x + 2) to obtain the quotient.  x2 + 11x + 10  = (x + 1) (x+10)  Here, -10, -1, -2 are the zeroes.  Hence, The answer is  -10, -1, -2
Relationship between Zeroes and Coefficients of a Polynomial

A fourth degree polynomial is called

A bi – quadratic polynomial

Quadratic polynomial

Cubic polynomial

Binomial

a

A polynomial of degree 2 is called a quadratic polynomial and a polynomial of degree 3 is called a cubic polynomial.

We know that, a polynomial of degree 2 is called a quadratic polynomial. A polynomial of degree 3 is called a cubic polynomial. A polynomial with 2 terms is called a binomial. This has nothing to do with the degree of the polynomial.  And, a polynomial of degree 4 is called a bi – quadratic polynomial. Hence, A fourth degree polynomial is called a bi – quadratic polynomial

Introduction to Polynomials

If the degree of the divisor g(x) is one then the degree of the remainder r (x) is.
1
3
2
0
d
The process of division is stopped when the remainder is zero or its degree is less than the degree of the divisor
We know that the process of division is stopped when the remainder is zero or its degree is less than the degree of the divisor. Hence, if the degree of the divisor is one, the degree of the remainder has to be 0. Hence, The answer is 0
Division Algorithm for Polynomials

If one of the zeros of the polynomial x3 + 2x2– 9x – 18x is -2, what are the other possible zeros?
6, 0
-3, 3
9, 0
-3, 2
b
Factorize the polynomial
Given, x3 + 2x2 – 9x – 18 has -2 as root x3 + 2x2 – 9x – 18x = x2(x+ 2) – 9(x+ 2) = (x2 – 9) (x+2) = (x + 3) (x – 3) (x + 2) [ using a2 – b2 = (a + b) (a – b)] Hence, possible zeroes of polynomial are -2, + 3 and -3. Thus, as -2 is already given other possible zeroes of given polynomial are +3 and -3.  Hence, The answer is -3,3
Relationship between Zeroes and Coefficients of a Polynomial

Which of the following polynomial when divided by (x + 1) given quotient 3x – 2 and leaves remainder 3?

3x2 + x + 1

X2 – x – 1

3x2 – x- 3

3x2 + x – 2

a

Dividend = Divisor × Quotient + Remainder

Here, divisor = (x + 1) Quotient = 3x – 2 Remainder = 3 We know that, dividend = divisor × quotient + remainder = (x + 1) × (3x – 2) + 3 = 3x2 + 3x – 2x – 2 + 3 = 3x2 + x + 1 Hence, The answer is 3x2 + x + 1

Division Algorithm for Polynomials

If p (x) is a polynomial of degree one and p(a) = 0, then a is said to be:

Value of p(x)

Zero of p(x)

Constant of p(x)

None of the above

b

Assume p (x) = mx + n

Let p (x) = mx + n Put x = a P(a) = ma + n = 0 a = -n/m Hence, The answer is zero of p(x)

Division Algorithm for Polynomials

If the discriminated of a quadratic polynomial, d > 0, then the polynomial has.

Two real and equal roots

Two real and unequal roots

Imaginary roots

No roots

b

Basic knowledge of polynomials

If the discriminated of a quadratic polynomial, d > 0, then the polynomial has two real and unequal roots. Hence, The answer is two real and unequal roots

Introduction to Polynomials

If x101 + 1001 is divided by x + 1, then remainder is
0
1
1490
1000
d
Given, x101 + 1001 P(x) is divided by x + 1 P (-1) = (-1101) + 1001  = 1000 Hence, The remainder is 1000
Division Algorithm for Polynomials

Which of the following is not a constant polynomial?

p(y) = y0

p(y) = x0

p(y) = y/y

p(x) = yx

d

x0 = 1

We know that if the power of any variable is zero then the answer is always 1.  For example, x0 = 1 which is a constant. Since p(x) = yx is a polynomial with variables x, y and there is no constant term in it. Hence, p(x) = yx is not a constant polynomial. Hence, The answer is p(x) = yx

Introduction to Polynomials

If ‘α’, ‘β’ and ‘ϒ’ are the zeroes of a cubic polynomial ax3 + bx2 + cx + d, then αβϒ =?

-d/a

-c/a

d/a

b/a

a

Product of zeroes of a cubic polynomial ax3 + bx2 + cx + d = - constant term / coefficient of x3

If ‘α’, ‘β’ and ‘ϒ’ are the zeroes of a cubic polynomial ax3 + bx2 + cx + d,  Since sum of the product of zeroes of a cubic polynomial ax3 + bx2 + cx + d = - constant term / coefficient of x3 Then, αβϒ = -d/a Hence, The value of αβϒ is -d/a

Relationship between Zeroes and Coefficients of a Polynomial

Which of the following numbers represents the sum of zeroes of the polynomial x2 – (√2 + √3) x + √6?

√6

√6 / (√2+ √3)

(√2+ √3) / √6

√2+ √3

d

Sum of zeroes = -b/a = - coefficient of x / coefficient of x2

Given, x2 – (√2+ √3) x + √6 Sum of zeroes = -b/a  = - coefficient of x / coefficient of x2 = - (√2+ √3) / 1 = √2+ √3 Hence, The answer is √2+ √3

Relationship between Zeroes and Coefficients of a Polynomial

Which of the following polynomials has the sum of its zeroes as ‘1’ and the product of its zeroes as -2?

x2 + x – 2

x2 + x + 2

x2 - x - 2

x2 + 2x + 1

c

Sum of zeroes = -b/a and product of zeroes = c/a

Since we know that, sum of zeroes = -b/a  = -(-1) / 1  = 1/1 = 1 Also, product of zeroes = c/a  = -2/1  = -2 So, the equation will be = ax2 + bx + c (1)x2 + (-1)x + (-2) = x2 - x – 2 Hence, The answer is x2 - x - 2

Relationship between Zeroes and Coefficients of a Polynomial

Which of the following expressions represents the remainder when x4 + x3 + 2x2 + 3x is divided by x2 + x?

5x

X

23x

4x

b

On dividing by x2 + x, we get (x2 + 2) + x / x2 + x

We have x4 + x3 + 2x2 + 3x   = x4 + 2x2 + x3 + 2x + x (break 3x as 2x + x) = x2 (x2 + 2) + x(x2 + 2) + x  = (x2 + x) (x2 + 2) + x On dividing by x2 + x, we get (x2 + 2) + x / x2 + x (it is of form Q + R/d) So, remainder is x. Hence, The remainder is x.

Division Algorithm for Polynomials

Write a quadratic polynomial, sum of whose zeroes is 2√3 and their product is 2.

x2-23x+2

x2-23x+1

x2-43x+2

x2-3x+1

a

x2 - (sum of the roots)x + (product of the roots)

Given sum of the roots =  2√3  product of the roots = 2 x2 - (sum of the roots)x + (product of the roots) x2 - (2√3)x + (2) Hence, The answer is x2-23x+2

Relationship between Zeroes and Coefficients of a Polynomial

Polynomials Level 4

3
5
-5
-3
a
Sum of both roots = -b/a and product of roots = c/a
Given, x2 – 4x + 1 Since we know that, sum of both roots = -b/a and product of roots = c/a Hence, The answer is 3
Relationship between Zeroes and Coefficients of a Polynomial

If graph of a polynomial does not intersect the x-axis but intersects y-axis in one point, then, no of zeroes of the polynomial is equal to

0

1

0 or 1

None of above

a

The number of zeroes of the polynomial is equal to the number of times graph intersect the x−axis

Since we know that, the number of zeroes of the polynomial is equal to the number of times graph intersect the x−axis. If the graph of the polynomial does not intersect x-axis, then the number of zeroes of the polynomial is zero. Hence, The answer is 0

Introduction to Polynomials

Which of the following graphs not be quadratic polynomials?

d

Zeros are equal to degree of a polynomial

Since we know that, zeros are equal to degree of a polynomial.  We can see that there are 3 zeroes in the graph (d) polynomial.  Therefore, it is not a quadratic polynomial. Hence, The answer is

Introduction to Polynomials

0
1
3
2
d
Degree of polynomial is equal to the zeros of the polynomial
Given, f(x) = (x - 2)2 + 4 Hence, The answer is 2
Intoduction to Polynomials

a

Hence, The answer is b - a + 1

Relationship between Zeroes and Coefficients of a Polynomial

none of above

a

If a > 0,  shape of parabola is open downwards and if a < 0 then the shape of parabola is open downwards

Hence, The answer is a > 0

Introduction to Polynomials

What is the number of zeroes that a linear polynomial has/ have?
1
2
3
4
a
Linear polynomial is a polynomial of degree one.
Linear Polynomials have only 1 zero. Hence, The answer is 1
Relationship between Zeroes and Coefficients of a Polynomial

What is the number of zeroes that a quadratic polynomial has/have?
1
2
3
4
b
A quadratic polynomial is a polynomial equation of a second degree
Since we know that, a quadratic polynomial is a polynomial equation of a second degree So, quadratic polynomial have only 2 zeroes. Hence, The answer is 2
Relationship between Zeroes and Coefficients of a Polynomial

b

Basic knowledge of polynomial

Given, -√5 and √5 are the zeroes of the quadratic polynomial. Hence, The answer is x2 - 5

Introduction to Polynomials

a

Equate the polynomial f(x) = 0 and solve

Given, f(x) = 2x - 5 Hence, The answer is (5/2,0)

Introduction to Polynomials

Both cannot positive

Both cannot negative

Both equal

Both unequal

a

Use Sridhar Acharya formula

Hence, The answer is both cannot be positive

Relationship between Zeroes and Coefficients of a Polynomial

24
14
-20
18
a
Given, f(x) = 3x2 – 5x + 2 and x = -2  Hence, The value of f(x) is 24
Introduction to Polynomials

-2
0
1
-1
c
Basic knowledge of polynomial
Given, (x + 1) is a factor of x2 – 3ax + 3a - 7 Hence, The value of a is 1
Factorization of Polynomials

1
-1
d
Given, S(x) = px2 + (p - 2)x + 2 Hence, The value of p is -1/3
Factorization of Polynomials

0
1
2
3
b
x3/ x2 = x
Hence, The answer is 1
Division Algorithm for Polynomials

0

1

2

Less than 2

d

Basic knowledge of polynomial

Given, x3 + 11 is divided by x2 - 3 The power of remainder is always less than divisor. Hence, The possible degree of remainder is less than 2

Division Algorithm for Polynomials

1
2
3
0
a
Substitute the value of x = 1 in the polynomial
Given, polynomial is x4 + 3x2 - 2x – 1 is divided by x - 1 Hence, The remainder is 1
Use of division algorithm of polynomial

-3
-2
-1
3
a
Substitute the value of x = -1 in the given polynomial
Hence, The remainder is -3
Division Algorithm for Polynomials

341
256
445
431
a
Basic knowledge of polynomial
Given, x2 – 11x + 30 Now, Hence, The value of a3 + b3 is 341
Introduction to Polynomials

a

Sum of zeroes = -b/a and product of zeroes = c/a

Hence, The third zero is -b/a

Relationship between Zeroes and Coefficients of a Polynomial

Has no linear term and the constant term is negative

Has no linear term and the constant term is positive

Can have a linear term but the constant term is negative

Can have a linear term but the constant term is positive

a

If one of the zeroes of quadratic polynomial p(x) is the negative  of the other  Hence, The answer is has no linear term and the constant term is negative

Let,

Introduction to Polynomials

If P(x) is a polynomial of degree one and P(a)=0, then a is said to be

Zero of P(x)

Value of P(x)

Constant of P(x)

None of above

a

Assume p(x) = mx + n and then substitute the value of x = a

Hence, The answer is zero of P(x)

Introduction to polynomials

None of above

a

Zeroes of the polynomial are 1, 1 + √2 and 1 – √2

Given, S(x) = x3 – 3x2 + x + 1 Also, the zeroes of the polynomial are (p - q), p, (p + q) Hence, The value of p and q are 1 , √2 or -√2

Relationship between Zeroes and Coefficients of a Polynomial

The polynomial with degree 6 is

x5 + y3 –x3 + x4

8x2y2z2 + 7x2y2z –x3 + x4

x5 + y3 –x3 + x7

x5 + y2 –x3 + x4

b

The highest power of a variable is called the degree of the polynomial

In the case of polynomials in more than one variable, the sum of the powers of the variables in each term is taken up and the highest sum so obtained is called the degree of the polynomial. 8x2y2z2 + 7x2y2z –x3 + x4 The highest power of this given expression is 2 + 2+ 2 = 6. Hence, The answer is 8x2y2z2 + 7x2y2z –x3 + x4

Degree of a Polynomial

Find the sum of all possible values of C for which the expression x3 + 3x2 - 9x + c can be expressed as the product of three factors, two of which are identical.
2
-2
20
-27
d
Let the roots of the equation, f(x) = x3 + 3x2 - 9x + c be α, α, β
Let the roots of the equation, f(x) = x3 + 3x2 - 9x + c be α, α, β f(x) = (x - α)2 (x - β) f’(x) = 3x2 + 6x – 9 = 0, has one root α f’(x) = x2 + 2x – 3 = 0 f’(x) = (x + 3)(x - 1) = 0 hence, α = -3 or 1 if α = 1, f(1) = 1 + 3 – 9 + c = 0 c – 5 = 0 c = 5 Similarly, if α = -3,  f(-3) = -27 + 27 + 27 + c = 0 c = -27 c = 5, -27 Hence, The answer is -27
Relationship between Zeroes and Coefficients of a Polynomial

The smallest integral value of a for which the equation x3 – x2 + ax – a = 0 have one exactly one roots, is
-1
0
1
2
c
Y’ = 3x2 – 2x + a ≥ 0 or ≤ 0
Given, y =  x3 – x2 + ax – a = 0 Y’ = 3x2 – 2x + a ≥ 0 or ≤ 0 Therefore d < 0 4 – 3.4.a < 0 1 – 3a < 0 a > 1/3, so 1 will be the answer  Hence, The answer is 1
Relationship between Zeroes and Coefficients of a Polynomial

A real number ‘k’ is said to be a zero of a polynomial p(x), if p (k) =?
3
2
0
1
c
If p(x) is a polynomial in x and k is any real number, then value of p(k) at  x = k is denoted by p (x) is found by replacing x by k in p (x).
A real number ‘k’ is said to be a zero of a polynomial p(x), if p (k) is equals to 0.  If p(x) is a polynomial in x and k is any real number, then value of p (k) at x = k is denoted by p (x) is found by replacing x by k in p (x). For example: In the polynomial x2 – 3x + 2, replacing x by 1 given, p (1) = 1 -3 + 2 = 0. Similarly, replacing x by 2 given, p (2) = 4 -6 + 2 = 0 for a polynomial p(x), real number k is said to be zero of polynomial p(x), if p (k) = 0. Hence, The answer is 0
Introduction to Polynomials

If the zeroes of a quadratic polynomial ax2 + bx + c, c≠ 0 are equal, then?

C and a have the same sign

B and c have opposite sign

B and c have the same sign

C and a have opposite sign

a

In the polynomial b2 is always positive and this is possible only if a and c both are either positive or negative

If the zeroes of a quadratic polynomial ax2 + bx + c, c≠ 0 are equal Then b2 – 4ac = 0 b2 = 4ac Here b2 is always positive and this is possible only if a and c both are either positive or negative.  both should have the same sign. Hence, C and a have the same sign

Introduction to Polynomials

The graph of a cubic polynomial x3 – 4x meets the x – axis at (-2, 0), (0, 0) and (2, 0), then the zeroes of the polynomial are

-2, 0 and 2

-2, 0 and 0

0, 0 and 2

None of these

a

Zeroes of the polynomial is the points on the graph of polynomial at which it passes through (or) touches the x- axis.

Since we know that, zeroes of the polynomial is the points on the graph of polynomial at which it passes through (or) touches the x- axis For finding the zeroes of the given polynomial, we will equate it with 0. So, x3 – 4x = 0 x(x2 - 4) = 0 we know that, (a2 – b2) = (a + b)(a - b) (x2 - 4) = (x2 - 22)  = (x + 2)(x - 2) Hence, the equation becomes: x(x + 2)(x - 2) = 0 Therefore, x = 0, x = 2, x = -2 So, the number zeroes are at -2, 0 and 2 (which are the coordinates of x – axis). Hence, The zeroes of the polynomial are -2, 0 and 2

Relationship between Zeroes and Coefficients of a Polynomial

The graph of the polynomial p(x) = ax – b, where a = 0; a, b ε R intersects x – axis at

(0, b/a)

(-b/a, 0)

(b/a, 0)

(-a/b, 0)

c

ax – b = 0

For intersecting at x – axis, p (x) = 0 as (y intersect = 0) ax – b = 0 x = b/a Hence, It intersects x-axis at (b/a, 0)

Introduction to Polynomials

The degree of the polynomial 4/5x2 – 7/5x + 2/3x3 + 6 is
2
1
3
0
c
Highest power in a polynomial p(x) is called degree of polynomial
Highest power in a polynomial p(x) is called degree of polynomial p(x),  p(x) = 4/5x2 – 7/5 x + 2/3x3 + 6 Since highest degree is 3. Hence, The highest degree of the given polynomial is 3.
Degree of a Polynomial

Which one of the following is the second degree polynomial function f (x) where f (0) = 5, f (-1) = 10 and f (1) = 6?

5x2 – 2x + 5

3x2 – 2x – 5

3x2 – 2x + 5

3x2 – 10x + 5

c

Assume  f(x) = a × x2 + b × x + c where a, b and c are real coefficients and a ≠ 0

Given, f (0) = 5, f (-1) = 10 and f (1) = 6 Let f(x) = a × x2 + b × x + c where a, b and c are real coefficients and a ≠ 0 Now, f(0) = c = 5 Also, f(-1) = a – b + 5 = 10        …(1) Similarly, f(1) = a + b + 5 = 6     ….(2) By adding (1) and (2), we get 2a + 10 = 16 a = 3 now, by substituting a = 3 in equation 1, we get 3 – b + 5 = 10 b = -2 Hence, f(x) = a × x2 + b × x + c = 3x2 – 2x + 5 Hence, The answer is 3x2 – 2x + 5

Relationship between Zeroes and Coefficients of a Polynomial

The degree of the polynomial x2 – 5x4 + 3/4x7 – 73x + 5 is_____

¾

4

7

-73

c

The degree of a polynomial is the highest power of its variable

Given equation is p (x) = x2 – 5x4 + 3/4x7 – 73x + 5  We know that, the degree of a polynomial is the highest power of its variable when the polynomial is expressed in its canonical form consisting of a linear combination of monomials. In the given equation, 7 is the highest power  Hence, The degree of polynomial p (x) is 7.

Degree of a Polynomial

The degree of (6x7 – 7x3 + 3x2 + 2x – 1) is____
6
3
7
5
c
The degree is the highest power of an exponent in the polynomial
Since the degree is the highest power of an exponent in the polynomial. In the given polynomial, (6x7 – 7x3 + 3x2 + 2x – 1) It can be clearly observed that the first term 6x7 has the highest power of x is 7. Thus, the degree of the given polynomial is 7.
Degree of a Polynomial

When x3 – 3x2 + 5x – 3 is divided by x2 – k, the remainder is 7x + a. then the value of k is________?
3
2
6
1
b
On long division, we have remainder = ( 5 + k)x - 3k – 3 = 7x + a equating the coefficient of x, we get 5 + k = 7  k = 7 - 5 k = 2 Hence, The value of k is 2
Division Algorithm for Polynomials

When the polynomial f (x) = 4x3 + 8x2 + 8x + 7 is divided by the polynomial g (x) = 2x2 – x + 1, the quotient and the remainder are

Quotient  = 2x + 5, remainder = 11x + 2

Quotient  = x – 5, remainder = 13

Quotient  = x2 – 5, remainder = 15

Quotient  = 2x – 5, remainder = 11x

a

Solve using long division method

Given, f (x) = 4x3 + 8x2 + 8x + 7 is divided by the polynomial g (x) = 2x2 – x + 1 By using the long division method, quotient = 2x + 5 and remainder = 11x + 2. Hence, The quotient and the remainder are 2x + 5 and 11x + 2 respectively

Division Algorithm for Polynomials

When x2 – 2x + k divides the polynomial x4 – 6x3 + 16x2 – 25x + 10, the remainder is ( x + a).  The value of a is ___
-5
3
5
-3
a
Use long division method and then solve further
If the polynomial x4 – 6x3 + 16x2 – 25x + 10 on division by x2 – 2x + k leaves remainder (x+ a), then, polynomial (x4 – 6x3 + 16x2 – 25x + 10) – (x+ a) on division by x2 – 2x + k leaves remainder zero. Using long division we get,  (x4 – 6x3 + 16x2 – 25x + 10) – (x+ a)  = x4 – 6x3 + 16x2 – 25x + 10 – a by x2 – 2x + k, the remainder obtained is (-10 + 2k)x + (10 –a – 8k + k2) So, (-10 + 2k)x + (10 –a – 8k + k2) = 0  = -10 + 2k = 0  and 10 – a – 8k + k2 = 0  k = 5 and a = -5  Hence, The value of a is -5
Division Algorithm for Polynomials

The value of ‘a’ so that (x+ 6) is a factor of the polynomial x3 + 5x2 – 4x + a is
0
12
13
10
b
If (x + 6) is a factor of the given polynomial, then the remainder should be zero.
If (x + 6) is a factor of the given polynomial, then the remainder should be zero. Dividing the polynomial by (x + 6) and equating the remainder, (a – 12) to 0, we get a – 12 = 0  or a = 12 Hence, The value of a is 12
Factorization of Polynomials

On dividing f (x) = x3 – 3x2 + x + 2 by a polynomial g(x), the quotient and remainder q (x) and r (x) are (x – 2) and (-2x + 4) respectively, then g(x) is

x2 + x -1

x2 + x -3

2x2 - x + 1

x2 - x +1

d

f (x) = g (x) q(x) + r(x)

Given, f (x) = x3 – 3x2 + x + 2, q(x) = (x – 2) and r(x) = (-2x + 4) We know that, f (x) = g (x) q(x) + r(x) to get g (x) By dividing f (x) – r (x) by q (x), we get g(x) = x2 - x +1 Hence, The g(x) os x2 - x + 1

Division Algorithm for Polynomials

If the points (5, 0), (1 -2) and (3, 6) lie on the graph of a polynomial. The zero of the polynomial is________

6

-2

5

Does not exist

c

The number of zeroes of p(x) is the number of times the curve intersects the x-axis, i.e., attains the value of 0.

Here, the polynomial p(x) meets the x –axis at 1 point.  (5, 0) is the point, where the graph cuts the x – axis. Therefore, x = 5 is the zero of the polynomial. Hence, The zero of the polynomial is 5

Introduction to Polynomials

P (x) = (k + 4) x2 + 13x + 3k is reciprocal of the other, and then the value of ‘k’ is
4
3
5
2
d
Product of roots α and 1/ α = c/a
Given, P (x) = (k + 4) x2 + 13x + 3k  Let one zero of the given polynomial be α The other zero is reciprocal = 1/ α Since product of roots α and 1/ α = c/a α × 1 / α = 3k / k + 4 1 = 3k / k + 4 k + 4 = 3k ( by cross multiplication) 4 = 3k –k 4 = 2k  k = 4/2 k = 2 Hence, The value of k is 2
Introduction to Polynomials