## Chapter 2 – Polynomials Exercise Ex. 2A

Find the zeros of the following quadratic polynomials and verify the relationship between the zeros and the coefficients: x^{2 }+ 7x + 12

Find the zeros of the following quadratic polynomials and verify the relationship

between the zeros and the coefficients: x^{2 }– 2x – 8

Find the zeros of the quadratic polynomial (x^{2} + 3x – 10) and verify the relation between its zeros and coefficients.

Question 4

Find the zeros of the quadratic polynomial (4x^{2} – 4x – 3) and verify the relation between its zeros and coefficients.

We have

Find the zeros of the quadratic polynomial (5x^{2} – 4 – 8x) and verify the relationship between its zeros and coefficients of the given polynomial.

Question 6

Find the zeros of the following quadratic polynomials and verify the relationship between the zeros and the coefficients:

Find the zeros of the quadratic polynomial (2x^{2} – 11x + 15) and verify the relation between its zeros and coefficients.

Question 8

between the zeros and the coefficients: 4x^{2 }– 4x + 1

Question 9

Find the zeros of the quadratic polynomial (x^{2} – 5) and verify the relation between its zeros and coefficients.

We have

So the zeros of f(x) are and

Question 10

Find the zeros of the quadratic polynomial (8x^{2} – 4) and verify the relation between its zeros and coefficients.

Let

Question 11

Find the zeros of the following quadratic polynomials and verify the relationship between the zeros and the coefficients: 5y^{2 }+ 10y

Question 12

Find the zeros of the following quadratic polynomials and verify the relationship between the zeros and the coefficients: 3x^{2 }– x – 4

Question 13

Find the quadratic polynomial whose zeros are 2 and -6. Verify the relation between the coefficients and the zeros of the polynomial.

Question 14

Find the quadratic polynomial whose zeros are . Verify the relation between the coefficients and the zeros of the polynomial.

Question 15

Find the quadratic polynomial, sum of whose zeros is 8 and their product is 12. Hence, find the zeros of the polynomial.

Question 16

Find the quadratic polynomial, sum of whose zeros is 0 and their product is -1. Hence, find the zeros of the polynomial.

Question 17

Find the quadratic polynomial, sum of whose zeros is and their product is 1. Hence, find the zeros of the polynomial.

Question 18

Question 19

If x =and x = -3 are the roots of the quadratic equation ax^{2 }+ 7x + b = 0 then find the values of a and b.

Question 20

If (x + a) is a factor of the polynomial 2x^{2} + 2ax + 5x + 10, find the value of a.

Question 21

One zero of the polynomial 3x^{3} + 16x^{2 }+ 15x – 18 is Find the other zeros of the polynomial.

## Chapter 2 – Polynomials Exercise Ex. 2B

Verify that 3, -2, 1 are the zeros of the cubic polynomial p(x) = x^{3} – 2x^{2} – 5x + 6 and verify the relation between its zeros and coefficients.

Verify that are the zeros of the cubic polynomial p(x) = 3x^{3} – 10x^{2} – 27x + 10 and verify the relation between its zeros and coefficients.

Question 3

Find a cubic polynomial whose zeros are 2, -3 and 4.

Question 4

Question 5

Find a cubic polynomial with the sum, sum of the product of its zeros taken two at a time, and the product of its zeros as 5, -2 and -24 respectively.

Question 6

Find the quotient and the remainder when: f(x) = x^{3} – 3x^{2 }+ 5x – 3 is divided by g(x)= x^{2} – 2.

Question 7

Find the quotient and the remainder when: f(x)= x^{4} -3x^{2} + 4x + 5 is divided by g(x)= x^{2} + 1 – x.

Question 8

Find the quotient and the remainder when: f(x)= x^{4} – 5x + 6 is divided by g(x) = 2 – x^{2}.

Question 9

By actual division, show that x^{2} – 3 is a factor of 2x^{4} + 3x^{3} – 2x^{2} – 9x – 12.

Question 10

On dividing 3x^{3} + x^{2} + 2x + 5 by a polynomial g(x), the quotient and remainder are (3x – 5) and (9x + 10) respectively. Find g(x).

Question 11

Verify division algorithm for the polynomials f(x) = 8 + 20x + x^{2} – 6x^{3 }and g(x) = 2 + 5x – 3x^{2}.

Question 12

It is given that -1 is one of the zeros of the polynomial x^{3} + 2x^{2} – 11x – 12. Find all the zeros of the given polynomial.

Question 13

If 1 and -2 are two zeros of the polynomial, find its third zero.

Question 14

Question 15

Question 16

Question 17

Question 18

Obtain all other zeros of , if two of its zeros are .

Find all the zeros of the polynomial , it being given that two of its zeros are .

## Chapter 2 – Polynomials Exercise Ex. 2C

If one zero of the polynomial x^{2} – 4x + 1 is (2 +), write the other zero.

Find

the zeros of the polynomial x^{2} + x – p(p

+ 1).

Find

the zeros of the polynomial x^{2} – 3x – m(m

+ 3).

Question 4

Question 5

If one zero of the quadratic polynomial kx^{2} + 3x + k is 2 then find

the value of k.

Question 6

If 3 is a zero of the polynomial 2x^{2} + x + k, find the value of k.

Question 7

If -4 is a zero of the polynomial x^{2 }– x – (2k + 2) then find the

value of k.

Question 8

If 1 is a zero of the polynomial ax^{2} – 3(a – 1)x – 1 then find the value of a.

Question 9

If -2 is a zero of the polynomial 3x^{2} + 4x + 2k then find the value of k.

Question 10

Write the zeros of the polynomial x^{2} – x – 6.

Question 11

If the sum of the zeros of the quadratic polynomial kx^{2} – 3x + 5 is 1, write the value of k.

Question 12

If the product of the zeros of the quadratic polynomial x^{2} – 4x + k is 3 then write the value of k.

Question 13

If (x + a) is a factor of (2x^{2} + 2ax + 5x + 10), find the value of a.

Question 14

If (a – b), a and (a + b) are zeros of the polynomial 2x^{3} – 6x^{2 }+ 5x – 7, write the value of a.

Question 15

If x^{3} + x^{2} – ax + b is divisible by (x^{2} – x), write the values of a and b.

Question 16

Question 17

State division algorithm for polynomials.

If f(x) and g(x) are any two polynomials with g(x) ≠ 0,

then we can always find polynomials q(x) and r(x) such that f(x) = q(x)g(x) + r(x),

where r(x) = 0

or degree r(x) < degree g(x).

Question 18

Question 19

Write the zeros of the quadratic polynomial f(x) = 6x^{2} – 3.

Question 20

Question 21

Question 22

Question 23

Question 24

Question 25

If the zeros of the polynomial f(x) = x^{3} – 3x^{2} + x + 1 are (a – b), a and (a + b), find a and b.

## Chapter 2 – Polynomials Exercise FA

Question 1

Zeros of p(x) = x^{2 }– 2x – 3 are

(a) 1, -3

(b) 3, -1

(c) -3, -1

(d)1, 3

Question 2

If 𝛼, 𝛽, 𝛾 are the zeros of the polynomial x^{3 }– 6x^{2 }– x + 30, then (𝛼𝛽 + 𝛽𝛾 + 𝛾𝛼) = ?

(a) -1

(b) 1

(c) -5

(d)30

Question 3

If 𝛼, 𝛽 are the zeros of kx^{2 }– 2x + 3k such that 𝛼 + 𝛽 = 𝛼𝛽, then k = ?

Question 4

It is given that the difference between the zeros of 4x^{2} – 8kx + 9 is 4 and k > 0. Then, k = ?

Question 5

Find the zeros of the polynomial x^{2 }+ 2x – 195.

Question 6

If one zeros of the polynomial (a^{2 }+ 9)x^{2 }+ 13x + 6a is the reciprocal of the other, find the value of a.

Question 7

Find a quadratic polynomial whose zeros are 2 and -5.

Question 8

If the zeros of the polynomial x^{3 }– 3x^{2 }+ x + 1 are (a – b), a and (a + b), find the values of a and b.

Question 9

Verify that 2 is a zero of the polynomial x^{3 }+ 4x^{2 }– 3x – 18.

Find the quadratic polynomial, the sum of whose zeros is -5 and their product is 6.

Question 11

Find a cubic polynomial whose zeros are 3, 5 and -2.

Question 12

Using remainder theorem, find the remainder

when p(x) = x^{3 }+ 3x^{2 }– 5x + 4 is divided by (x – 2).

Question 13

Show that (x + 2) is a factor of f(x) = x^{3 }+ 4x^{2 }+ x – 6.

Question 14

Question 15

If 𝛼, 𝛽 are the zeros of the polynomial f(x) = x^{2 }– 5x + k such that 𝛼– 𝛽 = 1, find the value of k.

Question 16

Show that the polynomial f(x) = x^{4 }+ 4x^{2 }+ 6 has no zero.

If one zero of the polynomial p(x) = x^{3 }– 6x^{2 }+ 11x – 6 is 3, find the other two zeros.

Question 18

Question 19

Find the quotient when p(x) = 3x^{4 }+ 5x^{3 }– 7x^{2 }+ 2x + 2 is divided by (x^{2 }+ 3x + 1).

Question 20

Use remainder theorem to find the value of

k, it being given that when x^{3 }+ 2x^{2 }+ kx + 3 is

divided by (x – 3), then the remainder is 21.

## Chapter 2 – Polynomials Exercise MCQ

Which of the following is a polynomial?

Correct answer: (d)

An expression of the form p(x) = a_{0} + a_{1}x + a_{2}x^{2} + ….. + a_{n}x^{n}, where a_{n} ≠ 0, is called a polynomial in x of degree n.

Here, a_{0}, a_{1}, a_{2}, ……, a_{n} are real numbers and each power of x is a non-negative integer.

Question 2

Which of the following is not a polynomial?

Correct answer: (d)

An expression of the form p(x) = a_{0} + a_{1}x + a_{2}x^{2} + ….. + a_{n}x^{n}, where a_{n} ≠ 0, is called a polynomial in x of degree n.

Here, a_{0}, a_{1}, a_{2}, ……, a_{n} are real numbers and each power of x is a non-negative integer.

Question 3

The zeros of the polynomial x^{2 }– 2x – 3 are

(a)-3, 1

(b)-3, -1

(c) 3, -1

(d) 3, 1

Question 4

Question 6

Question 7

The sum and the product of the zeros of a quadratic polynomial are 3 and -10 respectively. The quadratic polynomial is

(a) x^{2} – 3x + 10

(b) x^{2} + 3x – 10

(c) x^{2} – 3x – 10

(d) x^{2} + 3x + 10

Question 9

A quadratic polynomial whose zeros are 5 and -3, is

(a) x^{2} + 2x – 15

(b) x^{2} – 2x + 15

(c) x^{2} – 2x – 15

(d)none of these

Question 10

(a) 10x^{2} +x + 3

(b) 10x^{2} + x – 3

(c) 10x^{2} – x + 3

(d) 10x^{2} – x – 3

Question 11

The zeros of the quadratic polynomial x^{2 }+ 88x + 125 are

(a) both positive

(b) both negative

(c) one positive and one negative

(d) both equal

Question 12

If 𝛼 and 𝛽 are the zeroes of x^{2 }+ 5x + 8 then the value of (𝛼 + 𝛽) is

(a) 5

(b) -5

(c) 8

(d) -8

Question 13

If 𝛼 and 𝛽 are the zeros of 2x^{2 }+ 5x – 9 then the value of 𝛼𝛽 is

Question 14

If one zero of the quadratic polynomial kx^{2 }+ 3x + k is 2 then the value of k is

Question 15

If one zero of the quadratic polynomial (k – 1)x^{2 }+ kx + 1 is -4, then the value of k is

Question 16

If -2 and 3 are the zeros of the quadratic polynomial x^{2 }+ (a + 1)x + b then

(a) a = -2, b = 6

(b) a = 2, b = -6

(c) a = -2, b = -6

(d) a = 2, b = 6

Question 17

If one zero of 3x^{2} + 8x + k be the reciprocal of the other then k = ?

(a) 3

(b) -3

(c)

(d)

Question 18

If the sum of the zeros of the quadratic polynomial kx^{2 }+ 2x + 3k is equal to the product of its zeros then k = ?

Question 19

(a) 3

(b) -3

(c) 12

(d)-12

Question 20

If 𝛼, 𝛽, 𝛾 are the zeros of the polynomial x^{3 }– 6x^{2 }– x + 30, then (𝛼𝛽 + 𝛽𝛾 + 𝛾𝛼) = ?

(a) -1

(b) 1

(c) -5

(d)30

Question 21

If 𝛼, 𝛽, 𝛾 be the zeros of the polynomial 2x^{3 }+ x^{2 }– 13x + 6, then 𝛼𝛽𝛾

(a) -3

(b) 3

(c)

(d)

Question 22

If 𝛼, 𝛽, 𝛾 be the zeros of the polynomial p(x) such that (𝛼 + 𝛽 + 𝛾) = 3, (𝛼𝛽 + 𝛽𝛾 + 𝛾𝛼) = -10 and 𝛼𝛽𝛾 = -24, then p(x) =?

(a) x^{3 }+ 3x^{2 }– 10x + 24

(b) x^{3 }+ 3x^{2 }+ 10x – 24

(c) x^{3 }– 3x^{2 }– 10x + 24

(d) None of these

Question 23

If two of the zeros of the cubic polynomial ax^{3 }+ bx^{2 }+ cx + d are 0, then the third zero is

Question 24

If one of the zeros of the cubic polynomial ax^{3 }+ bx^{2 }+ cx + d is 0, then the product of other two zeros are

Question 25

If one of the zeros of the cubic polynomial x^{3 }+ ax^{2 }+ bx + c is -1, then the product of the other two zeros is

(a) a – b – 1

(b) b – a – 1

(c) 1 – a + b

(d) 1 + a – b

(a) 3

(b) -3

(c) -2

(d) 2

Question 27

On dividing a polynomial p(x) by a non-zero polynomial q(x), let g(x) be the quotient and r(x) be the remainder, then p(x) = q(x).g(x) + r(x), where

(a)r(x) = 0 always

(b)deg r(x) < deg g(x) always

(c) either r(x) = 0 or deg r(x) < deg g(x)

(d) r(x) = g(x)

Question 28

Which of the following is a true statement?

(a)x^{2 }+ 5x – 3 is a linear polynomial

(b)x^{2 }+ 4x – 1 is a binomial

(c) x + 1 is a monomial

(d) 5x^{3} is a monomial