# Solution of Chapter 2. Polynomials (RD Sharma - Mathematics Book)

## Exercise 2.1

1

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

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

2

If α and β are the zeros of the quadratic polynomial, find the value of

3

If α and β are the zeros of the quadratic polynomial, find the value of .

4

If α and β are the zeros of the quadratic polynomial , find the value of

.

5

If α and β are the zeros of the quadratic polynomial , find the value of .

6

If α and β are the zeros of the quadratic polynomial , find the value of .

7

If α and β are the zeros of the quadratic polynomial , find the value of .

8

If α and β are the zeros of the quadratic polynomial , find the value of

.

9

If α and β are the zeros of the quadratic polynomial , find the value of .

10

If one zero of the quadratic polynomial is negative of the other, find the value of k.

11

If the sum of the zeros of the quadratic polynomial is equal to their product, find the value of k.

12

If the squared difference of the zeros of the quadratic polynomial is equal to 144, find the value of p.

13

If α and β are the zeros of the quadratic polynomial , prove that .

14

If α and β are the zeros of the quadratic polynomial , show that .

15

If α and β are the zeros of the quadratic polynomial such that and, find a quadratic polynomial having α and β as its zeros.

16

If α and β are the zeros of the quadratic polynomial , find a quadratic polynomial whose zeros are .

17

If α and β are the zeros of the quadratic polynomial , find a quadratic polynomial whose zeros are .

18

If α and β are the zeros of the quadratic polynomial , from a polynomial whose zeros are .

19

If α and β are the zeros of the quadratic polynomial , find a polynomial whose roots are (i) (ii) .

20

If α and β are the zeros of the quadratic polynomial , then evaluate:

(i) (ii)

(iii) (iv)

(v) (vi)

(vii) (viii)

## Exercise 2.2

1

Verify that the numbers given along side of the cubic polynomials below are their zeros. Also, verify the relationship between the zeros and coefficients in each case:

(i)

(ii)

2

Find a cubic polynomial with the sum, sum of the product of its zeros taken two at a time, and product of its zeros as 3, -1 and -3 respectively.

3

If the zeros of the polynomial are in A.P., find them.

4

Find the condition that the zeros of the polynomial may be in A.P.

5

If the zeros of the polynomial are in A.P., prove that .

6

If the zeros of the polynomial are in A.P., find the value of k.

## Exercise 2.3

1

Apply division algorithm to find the quotient q(x) and remainder r(x) on dividing f(x) by g(x) in each of the following:

(i)

(ii)

(iii)

(iv)

2

Check whether the first polynomial is a factor of the second polynomial by applying the division algorithm:

(i)

(ii)

(iii)

3

Obtain all zeros of the polynomial , if two of its zeros are -2 and -1.

4

Obtain all zeros of if one of its zeros is -2.

5

Obtain all zeros of the polynomial if two of its zeros are and .

6

Find all zeros of the polynomial it its two zeros are and .

7

What must be added to the polynomial so that the resulting polynomial is exactly divisible by ?

8

What must be subtracted from the polynomial so that the resulting polynomial is exactly divisible by ?

9

Find all the zeros of the polynomial , if two of its zeros are 2 and-2.

10

Find all zeros of the polynomial , if two of its zeros are and .

11

Find all the zeros of the polynomial , if two of its zeros are and .

12

Find all the zeros of the polynomial , if two of its zeros are and .

## CCE - Formative Assessment

1

Define a polynomial with real coefficients.

2

Define degree of a polynomial.

3

Write the standard form of a linear polynomial with real coefficients.

4

Write the standard form of a quadratic polynomial with real coefficients.

5

Write the standard form of a cubic polynomial with real coefficients.

6

Define the value of a polynomial at a point.

7

Define zero of a polynomial.

8

The sum and product of the zeros of a quadratic polynomial are and — 3 respectively. What is the quadratic polynomial?

9

Write the family of quadratic polynomials having and 1 as its zeros.

10

If the product of zeros of the quadratic polynomial f(x) = x2 — 4x + k is 3, find the value of k.

11

If the sum of the zeros of the quadratic polynomial f (x) = kx2 — 3x + 5 is 1, write the value of k.

12

In Fig. 2.17, the graph of a polynomial p(x) is given. Find the zeros of the polynomial.

13

The graph of a polynomial y = f (x), shown in Fig. 2.18. Find the number of real zeros of f (x).

14

The graph of the polynomial f(x) = ax2 + bx + c is as shown below (Fig. 2.19). Write the signs of 'a' and b2 – 4ac.

15

The graph of the polynomial f(x) = ax2 + box + c is as shown in Fig. 2.20. Write the value of b2 – 4ac and the number of real zeros of f (x).

16

In Q. No. 14, write the sign of c.

17

In Q. No. 15, write the sign of c.

18

The graph of a polynomial/ (x) is as shown in Fig. 2.21. Write the number of real zeros of f (x).

19

If x = 1 is a zero of the polynomial f (x) = x3 – 2x2 + 4x + k, write the value of k.

20

State division algorithm for polynomials.

21

Give an example of polynomials f(x), g(x), q(x) and r(x) satisfying f(x) = g(x) .q(x)+ r(x), where degree r (x) = 0.

22

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

23

If fourth degree polynomial is divided by a quadratic polynomial, write the degree of the remainder.

24

If f(x) = x3 + x2 – ax + b is divisible by x2 – x write the values of a and b.

25

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

26

Write the coefficients of the polynomial p(z) = z5 – 2z2 + 4.

27

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

28

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

29

For what value of k, – 4 is a zero of the polynomial x2 – x – (2k + 2)?

30

If 1 is a zero of the polynomial p(x) = ax2 – 3(a – 1) x – 1, then find the value of a.

31

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

32

If α, β are the zeros of the polynomial 2y2 + 7y + 5, write the value of α + β + αβ.

33

For what value of k, is 3 a zero of the polynomial 2x2 + x + k?

34

For what value of k, is – 3 a zero of the polynomial x2 + 11x + k?

35

For what value of k, is – 2 a zero of the polynomial 3x2 + 4x + 2k?

36

If a quadratic polynomial f (x) is factorizable into linear distinct factors, then what is the total number of real and distinct zeros of f(x)?

37

If a quadratic polynomial f (x) is a square of a linear polynomial, then its two zeroes are coincident. (True/False)

38

If a quadratic polynomial f(x) is not factorizable into linear factors, then it has no real zero. (True/False)

39

If f(x) is a polynomial such that f(a)f(b)< 0, then what is the number of zeros lying between a and b?

40

If graph of quadratic polynomial ax2 + bx + c cuts positive direction of y-axis, then what is the sign of c?

41

If the graph of quadratic polynomial ax2 + bx + c cuts negative direction of y-axis, then what is the sign of c?

1

If α, β are the zeros of the polynomial f (x) = x2 + x + 1, then =

2

If α, β are the zeros of the polynomial p(x) = 4x2 + 3x + 7, then =

3

If one zero of the polynomial f (x) = (k2 + 4) x2 + 13x + 4k is reciprocal of the other, then k =

4

If the sum of the zeros of the polynomial f(x) = 2x3 – 3kx2 + 4x – 5 is 6, then the value of k is

5

If α and β are the zeros of the polynomial f(x) = x2 + px + q, then a polynomial having and is its zeros is

6

If α, β are the zeros of polynomial f(x) = x2 – p (x + 1) – c, then (α + 1) (β + 1) =

7

If α, β are the zeros of the polynomial f(x) = x2 – p (x + 1) – c such that (α + 1) (β + 1) = 0, then c =

8

If f(x) = ax2 + bx + c has no real zeros and a + b + c < 0, then

9

If the diagram in Fig. 2.22 shows the graph of the polynomial f (x) = ax2 + bx + c, then

10

Figure 2.23 shows the graph of the polynomial f(x) = ax2 + bx + c for which

11

If the product of zeros of the polynomial f(x) = ax3 –6x2 + 11x – 6 is 4, then a =

12

If zeros of the polynomial f (x) = x3 – 3px2 + qx – r are in A.P., then

13

If the product of two zeros of the polynomial f (x) = 2xZ3 + 6x2 – 4x + 9 is 3, then its third zero is

14

If the polynomial f(x) = ax3 + bx – c is divisible by the polynomial g(x) = x2 + bx + c, then ab =

15

In Q. No. 14, c =

16

If one root of the polynomial f (x) = 5x2 + 13x + k is reciprocal of the other, then the value of k is

17

If α, β, γ are the zeros of the polynomial f(x) = ax3 + bx2 + cx + d, then =

18

If α, β, γ are the zeros of the polynomial f(x) = ax3 + bx2 + cx + d, then α2 + β2 + γ2 =

19

If α, β, γ are the zeros of the polynomial f(x) = x3 – px2 + qx – r, then =

20

If α, β are the zeros of the polynomial f(x) = ax2 + bx + c, then =

21

If two of the zeros of the cubic polynomial ax3 + bx2 + cx + d are each equal to zero, then the third zero is

22

If two zeros of x3 + x2 – 5x – 5 are √5 and –√5 , then its third zero is

23

The product of the zeros of x3 + 4x2 + x – 6 is

24

What should be added to the polynomial x2 – 5x + 4, so that 3 is the zero of the resulting polynomial?

25

What should be subtracted to the polynomial x2 – 16x + 30, so that 15 is the zero of the resulting polynomial?

26

A quadratic polynomial, the sum of whose zeroes is 0 and one zero is 3, is

27

If two zeroes of the polynomial x3 + x2 – 9x – 9 are 3 and – 3, then its third zero is

28

If √5 and – √5 are two zeroes of the polynomial x3 + 3x2 – 5x – 15, then its third zero is