# Solution of Chapter 2. Polynomials (RS Aggarwal - Mathematics Book)

## Exercise 2A

1

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

x2 + 7x + 12

2

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

x2 + 2x – 8

3

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

x2 + 3x – 10

4

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

4x2 – 4x – 3

5

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

5x2 – 4 – 8x

6

Find the zeros of the following quadratic polynomials and verify the relationship between the zeros and the coefficients: 7

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

2x2 – 11x + 15

8

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

4x2 – 4x + 1

9

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

x2 – 5

10

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

8x2 – 4

11

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

5y2 + 10y

12

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

3x2 – x – 4

13

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

14

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

15

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

16

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

17

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

18

Find the quadratic polynomial, the sum of whose roots is √2 and their product is 1/3.

19

If and x = ‒3 are the roots of the quadratic equation ax2 + 7x + b = 0 then find the values of a and b.

20

If (x + a) is a factor of the polynomial 2x2 + 2ax + 5x + 10, find the value of a.

21

One zero of the polynomial 3x3 + 16x2 + 15x – 18 is 2/3. Find the other zeros of the polynomial.

## Exercise 2B

1

Verify that 3, ‒2, 1 are the zeros of the cubic polynomial p(x) = x3 – 2x2 – 5x + 6 and verify the relation between its zeros and coefficients.

2

Verify that 5, –2 and are the zeros of the cubic polynomial p(x) = 3x3 - 10x2– 27x + 10 and verify the relation between its zeros and coefficients

3

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

4

Find a cubic polynomial whose zeros are , 1 and –3.

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.

6

Find the quotient and the remainder when:

f(x) = x3 – 3x2 + 5x –3 is divided by g(x) = x2 – 2.

7

Find the quotient and the remainder when:

f(x) = x4 – 3x2 + 4x + 5 is divided by g(x) = x2 + 1 – x.

8

Find the quotient and the remainder when:

f(x) = x4 – 5x + 6 is divided by g(x) = 2 – x2.

9

By actual division, show that x3 – 3 is a factor 2x4 + 3x3 – 2x2 – 9x – 12.

10

On dividing 3x3 + x2 + 2x + 5 by a polynomial g(X), the quotient and remainder are (3x – 5) and (9x + 10) respectively. Find g(x).

11

Verify division algorithm for the polynomials f(x) = 8 + 20x + x2 ‒ 6x3 and g(x) = 2 + 5x ‒ 3x2.

12

It is given that ‒1 is one of the zeros of the polynomial x3 + 2x2 ‒ 11x ‒ 12. Find all the zeros of the given polynomial.

13

If 1 and ‒2 are two zeros of the polynomial (x3 ‒ 4x2 ‒ 7x + 10), find its third zero.

14

If 3 and ‒3 are two zeros of the polynomial (x4 + x3 ‒ 11x2 ‒ 9x + 18), find all the zeros of the given polynomial.

15

If 2 and ‒2 are two zeros of the polynomial (X4 + x3 ‒ 34x2 ‒ 4x + 120), find all the zeros of the given polynomial.

16

Find all the zeros of (x4 + x3 ‒ 23x2 ‒ 3x + 60), if it is given that two of its zeros are √3 and ‒√3.

17

Find all the zeros of (2x4 ‒ 3x3 ‒ 5x2 + 9x ‒ 3), it being given that two of its zeros are √3 and – √3.

18

Obtain all other zeros of (x4 + 4x3 ‒ 2x2 ‒ 20x ‒15) if two of its zeros are √5 and – √5.

19

Find all the zeros of the polynomial (2x4 ‒ 11x3 + 7x2 + 13x ‒ 7), it being given that two of its zeros are (3 + √3) and (3 ‒ √3)

## Exercise 2C

1

If one zero of the polynomial x2 ‒ 4x + 1 is (2 + √3), write the other zero.

2

Find the zeros of the polynomial x2 + x ‒ p (p + 1).

3

Find the zeros of the polynomial x2 – 3x – m (m + 3).

4

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

5

If one zeros of the quadratic polynomial kx2 + 3x + k is 2 then find the value of k.

6

If 3 is a zero of the polynomial 2s2 + x + k, find the value of k.

7

If ‒4 is a zero of the polynomial x2 ‒ x (2k + 2) then find the value of k.

8

If 1 is a zero of the polynomial ax2 ‒ 3 (a ‒ 1) x ‒ 1 then find the value of a.

9

If ‒2 is a zero of the polynomial 3x2 + 4x + 2k then find the value of k.

10

Write the zeros of the polynomial x2 ‒ x ‒ 6.

11

If the sum of the zeros of the quadratic polynomial kx2 ‒ 3x + 5 is 1, write the value of k.

12

If the product of the zeros of the quadratic polynomial x2 ‒ 4x + k is 3 then write the value of k.

13

If (x + a) is a factor of (2x2 + 2ax + 5x + 10), find the value of a.

14

If (a ‒ b), a and (a + b) are zeros of the polynomial 2x3 ‒ 6x2 + 5x – 7, write the value of a.

15

If x3 + x2 ‒ ax + b is divisible by (x2 ‒ x), write the values of a and b.

16

If α and β are the zeros of the polynomial 2x2 + 7x + 5, write the value of α + β + αβ.

17

State division algorithm for polynomials.

18

The sum of the zeros and the product of zeros of a quadratic polynomial are -1/2 and ‒3 respectively. Write the polynomial.

19

Write the zeros of the quadratic polynomial f(X) = 6x2 ‒ 3.

20

Write the zeros of the quadratic polynomial f(x) = 4√3x2 + 5x ‒ 2√3.

21

If α and β are the zeros of the polynomial f(x) = x2 ‒ 5x + k such that α ‒ β = 1, find the value of k.

22

If α and β are the zeros of the polynomial f(x) = 6x2 + x ‒ 2, find the value of 23

If α and β are the zeros of the polynomial f(x) = 5x2 ‒ 7x + 1, find the value of 24

If α and β are the zeros of the polynomial f(x) = x2 + x – 2, find the value of 25

If the zeros of the polynomial f(x) = x3 – 3x2 + x + 1 are (a – b), a and (A + b), find the a and b.

## Multiple Choice Questions (MCQ)

1

Which of the following is a polynomial?

2

Which of the following is not a polynomial?

3

The zeros of the polynomial x2 – 2x – 3 are

4

The zeros of the polynomial x2 ‒ √2 x – 12 are

5

The zeros of the polynomial 4x2 + 5√2x – 3 are

6

The zeros of the polynomial are

7

The zeros of the polynomial are

8

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

9

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

10

A quadratic polynomial whose zeros are , is

11

The zeros of the quadratic polynomial x2 + 88x + 125 are

12

If α and β are the zeros of x2 + 5x + 8 then the value of (α + β) is

13

If α and β are the zeros of 2x2 + 5x – 9 then the value of αβ is

14

If one zero of the quadratic polynomial kx2 + 3x + k is 2 then the value of k is

15

If one zero of the quadratic polynomial (k – 1) x2 + kx + 1 is –4 then the value of k is

16

If –2 and 3 are the zeros of the quadratic polynomial x2 + (a + 1) x + b then

17

If one zero of 3x2 + 8x + k be the reciprocal of the other then k = ?

18

If the sum of the zeros of the quadratic polynomial kx2 + 2x + 3k is equal

19

If α, β are the zeros of the polynomial x2 + 6x + 2 then ?

20

If α, β, γ are the zeros of the polynomial x3 – 6x2 – x + 30 then (αβ + βγ + γ α) = ?

21

If α, β, γ are the zeros of the polynomial 2x3 + x2 – 13x + 6 then αβγ = ?

22

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

23

If two of the zeros of the cubic polynomial az3 + bx2 + cx + d are 0 then in the third zero is

24

If one of the zeros of the cubic polynomial ax3 + bx2 + cx + d is 0 then the product of the other two zeros is

25

If one of the zeros of the cubic polynomial x3 + ax2 + bx + c is –1 then the product of the other two zeros is

26

If α, β be the zeros of the polynomial 2x2 + 5x + k such that then k = ?

27

One dividing a polynomial p(x) by a nonzero 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

28

Which of the following is a true statement?

## Formative Assessment (Unit Test)

1

Zeros of p(x) = x2 – 2x – 3 are

2

If α, β, γ are the zeros of the polynomial x3 – 6x2 – x + 30 then the value of (αβ + βγ + γ α) is

3

If α, β are the zeros of kx2 – 2x + 3k such that α + β = αβ then k = ?

4

If is given that the difference between the zeros of 4x2 – 8kx + 9 is 4 and k > 0. Then k = ?

5

Find the zeros of the polynomial x2 + 2x – 195.

6

If one zero of the polynomial (a2 + 9) x2 + 13x + 6a is the reciprocal of the other, find the value of a.

7

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

8

If the zeros of the polynomial x3 – 3x2 + x + 1 are (a ‒ b), a and (a + b), find the values of a and b

9

Verify that 2 is a zero of the polynomial x3 + 4x2– 3x – 18.

10

Find the quadratic polynomial, the sum of whose zeros is ‒5 and their products is 6.

11

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

12

Using remainder theorem, find the remainder when p(x) = x3 + 3x2 – 5x + 4 is divided by (x ‒ 2).

13

Show that (x + 2) is a factor of f(x) = x3 + 4x2 + x – 6.

14

If α, β, γ are the zeros of the polynomial p(x) = 6x3 + 3x2 – 5x + 1, find the value of .

15

If α, β are the zeros of the polynomial f(x) = x2 – 5x + k such that α ‒ β = 1, find the value of k.

16

Show that the polynomial f(x) = x4 + 4x2 + 6 has no zero.

17

If one zero of the polynomial p(x) = x3 – 6x2 + 11x – 6 is 3, find the other two zeros.

18

If two zeros of the polynomial p(x) = 2x4 – 3x3 – 3x2 + 6x – 2 are √2 and – √2, find its other two zeros.