# RD Sharma Solutions for CBSE Class 10 Mathematics chapter 2 - Polynomials

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## Chapter 2 - Polynomials Excercise Ex. 2.1

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

f(x) = x^{2}
- 2x - 8

x^{2} - 2x - 8
= x^{2} - 4x + 2x - 8
= x(x - 4) + 2(x - 4)
= (x - 4)(x + 2)
The zeroes of the quadratic equation are 4 and -2.
Let ∝ = 4 and β
= -2
Consider f(x) = x^{2} - 2x - 8
Sum of the zeroes = …(i)
Also, ∝ + β = 4 -
2 = 2 …(ii)
Product of the zeroes = …(iii)
Also, ∝ β = -8 …(iv)
Hence, from (i), (ii), (iii) and (iv),the relationship
between the zeroes and their coefficients isverified.

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

g(s) = 4s^{2}
- 4s + 1

4s^{2}
- 4s + 1
= 4s^{2} - 2s - 2s + 1
= 2s(2s - 1) - (2s - 1)
= (2s - 1)(2s - 1)
The zeroes of the quadratic equation are and .
Let ∝ = and β =
Consider4s^{2} - 4s + 1
Sum of the zeroes = …(i)
Also, ∝ + β = …(ii)
Product of the zeroes = …(iii)
Also, ∝ β = …(iv)
Hence, from (i), (ii), (iii) and (iv),the relationship
between the zeroes and their coefficients is verified.

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

h (t) = t^{2}
- 15

h (t) = t^{2} - 15 = (t + √15)(t - √15)
The zeroes of the quadratic equation areand .
Let ∝ = and β =
Considert^{2} - 15 = t^{2} - 0t - 15
Sum of the zeroes = …(i)
Also, ∝ + β = …(ii)
Product of the zeroes = …(iii)
Also, ∝ β = …(iv)
Hence, from (i), (ii), (iii) and (iv),the relationship
between the zeroes and their coefficients is verified.

f(s) = 6x^{2}
- 3 - 7x

f(s) = 0
6x^{2} - 3 - 7x =0 6x^{2} - 9x + 2x - 3 = 0
3x (2x - 3) + (2x - 3) = 0
(3x + 1) (2x - 3) = 0
The zeroes of a quadratic equation are and
.
Let ∝ = and β =
Consider6x^{2} - 7x - 3 = 0
Sum of the zeroes = …(i)
Also, ∝ + β = …(ii)
Product of the zeroes = …(iii)
Also, ∝ β = …(iv)
Hence, from (i), (ii), (iii) and (iv),the relationship
between the zeroes and their coefficients is verified.

The zeroes of a quadratic equation are and . Let ∝ = and β = Consider Sum of the zeroes = …(i) Also, ∝ + β = …(ii) Product of the zeroes = …(iii) Also, ∝ β = …(iv) Hence, from (i), (ii), (iii) and (iv),the relationship between the zeroes and their coefficients is verified.

The zeroes of a quadratic equation are and . Let ∝ = and β = Consider Sum of the zeroes = …(i) Also, ∝ + β = …(ii) Product of the zeroes = …(iii) Also, ∝ β = …(iv) Hence, from (i), (ii), (iii) and (iv),the relationship between the zeroes and their coefficients is verified.

The zeroes of a quadratic equation are and 1. Let ∝ = and β = 1 Consider Sum of the zeroes = …(i) Also, ∝ + β = …(ii) Product of the zeroes = …(iii) Also, ∝ β = …(iv) Hence, from (i), (ii), (iii) and (iv),the relationship between the zeroes and their coefficients is verified.

The zeroes of a quadratic equation are a and. Let ∝ = a and β = Consider Sum of the zeroes = …(i) Also, ∝ + β = …(ii) Product of the zeroes = …(iii) Also, ∝ β = …(iv) Hence, from (i), (ii), (iii) and (iv),the relationship between the zeroes and their coefficients is verified.

The zeroes of a quadratic equation are and . Let ∝ = and β = Consider Sum of the zeroes = …(i) Also, ∝ + β = …(ii) Product of the zeroes = …(iii) Also, ∝ β = …(iv) Hence, from (i), (ii), (iii) and (iv),the relationship between the zeroes and their coefficients is verified.

The zeroes of a quadratic equation are and . Let ∝ = and β = Consider=0 Sum of the zeroes = …(i) Also, ∝ + β = …(ii) Product of the zeroes = …(iii) Also, ∝ β = …(iv) Hence, from (i), (ii), (iii) and (iv),the relationship between the zeroes and their coefficients is verified.

The zeroes of a quadratic equation are and. Let ∝ = and β = Consider=0 Sum of the zeroes = …(i) Also, ∝ + β = …(ii) Product of the zeroes = …(iii) Also, ∝ β = …(iv) Hence, from (i), (ii), (iii) and (iv),the relationship between the zeroes and their coefficients is verified.

The zeroes of a quadratic equation are and. Let ∝ = and β = Consider=0 Sum of the zeroes = …(i) Also, ∝ + β = …(ii) Product of the zeroes = …(iii) Also, ∝ β = …(iv) Hence, from (i), (ii), (iii) and (iv),the relationship between the zeroes and their coefficients is verified.

For the following, find a quadratic polynomial whose sum and product respectively of the zeroes are as given. Also, find the zeroes of these polynomials by factorization.

For the following, find a quadratic polynomial whose sum and product respectively of the zeroes are as given. Also, find the zeroes of these polynomials by factorization.

For the following, find a quadratic polynomial whose sum and product respectively of the zeroes are as given. Also, find the zeroes of these polynomials by factorization.

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

## Chapter 2 - Polynomials Excercise Ex. 2.2

On comparing the given polynomial with the polynomial ax

^{3}+ bx

^{2}+ cx + d, we obtain a = 2, b = 1, c = -5, d = 2

Thus, the relationship between the zeroes and the coefficients is verified.

On comparing the given polynomial with the polynomial ax

^{3}+ bx

_{2}+ cx + d, we obtain a = 1, b = -4, c = 5, d = -2.

Thus, the relationship between the zeroes and the coefficients is verified.

**Concept insight:**The zero of a polynomial is that value of the variable which makes the polynomial 0. Remember that there are three relationships between the zeroes of a cubic polynomial and its coefficients which involve the sum of zeroes, product of all zeroes and the product of zeroes taken two at a time.

^{3}- 15x

^{2}+ 37x - 30 are in A.P., find them.

## Chapter 2 - Polynomials Excercise Ex. 2.3

## Chapter 2 - Polynomials Excercise 2.61

(a) 1

(b) -1

(c) 0

(d) None of these

So, the correct option is (d).

If one zero of the polynomial f(x) = (k^{2} + 4)x^{2 }+ 13x + 4k is reciprocal of the other, then k =

(a) 2

(b) -2

(c) 1

(d) -1

## Chapter 2 - Polynomials Excercise 2.62

If the sum of the zeros of the polynomial f(x) = 2x^{3} - 3kx^{2} + 4x - 5 is 6, then value of k is

(a) 2

(b) 4

(c) -2

(d) -4

So, the correct option is (b).

(a) x^{2 }+ qx + p

(b) x^{2} - px + q

(c) qx^{2} + px + 1

(d) px^{2} + qx + 1

If α, β are the zeros of polynomial f(x) = x^{2 }- p(x + 1) - c, then (α + 1) (β + 1) =

(a) c - 1

(b) 1 - c

(c) c

(d) 1 + c

If α, β are the zeros of the polynomial f(x) = x^{2 }- p(x + 1) - c such that (α + 1) (β + 1) = 0 then c =

(a) 1

(b) 0

(c) -1

(d) 2

Given that (α + 1) (β + 1) = 0

So, the correct option is (a).

If f(x) = ax^{2} + bx + c has no real zeros and a + b + c < 0, then

(a) c = 0

(b) c > 0

(c) c < 0

(d) None of these

We know that, if the quadratic equation ax^{2} + bx + c = 0 has no real zeros

then

Case 1:

a > 0, the graph of quadratic equation should not intersect x - axis

It must be of the type

Case 2 :

a < 0, the graph will not intersect x - axis and it must be of type

According to the question,

a + b + c < 0

This means,

f(1) = a + b + c

f(1) < 0

Hence, f(0) < 0 [as Case 2 will be applicable]

So, the correct option is (c).

If the diagram in figure show the graph of the polynomial f(x) = ax^{2} + bx + c then

(a) a > 0, b < 0 and c > 0

(b) a < 0, b < 0 and c < 0

(c) a < 0, b > 0 and c > 0

(d) a < 0, b > 0 and c < 0

Figure shows the graph of the polynomial f(x) = ax^{2} + bx + c for which

(a) a < 0, b > 0 and c > 0

(b) a < 0, b < 0 and c > 0

(c) a < 0, b < 0 and c < 0

(d) a > 0, b > 0 and c < 0

If zeros of the polynomial f(x) = x^{3}^{ }- 3px^{2} + qx - r are in A.P, then

(a) 2p^{3} = pq - r

(b) 2p^{3} = pq + r

(c) p^{3} = pq - r

(d) None of these

## Chapter 2 - Polynomials Excercise 2.63

So, the correct option is (b).

In Q. No. 14, c =

(a) b

(b) 2b

(c) 2b^{2}

(d) -2b

So, the correct option is (c).

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

The product of the zeros of x^{3 }+ 4x^{2} + x - 6 is

(a) -4

(b) 4

(c) 6

(d) -6

## Chapter 2 - Polynomials Excercise 2.64

What should be added to the polynomial x^{2} - 5x + 4, so that 3 is the zero of the resulting polynomial ?

(a) 1

(b) 2

(c) 4

(d) 5

We know that, if α and β are roots of ax^{2 }+ bx + c = 0 then they must satisfy the equation.

According to the question, the equation is

x^{2 }- 5x + 4 = 0

If 3 is the root of equation it must satisfy equation.

x^{2 }- 5x + 4 = 0

but f(3) = 3^{2 }- 5(3) + 4 = -2

so, 2 has to be added in the equation.

So, the correct option is (b).

What should be subtracted to the polynomial x^{2}^{ }- 16x + 30, so that 15 is the zero of resulting polynomial?

(a) 30

(b) 14

(c) 15

(d) 16

We know that, if α and β are roots of ax^{2 } + bx + c = 0, then α and β must satisfy the equation.

According to the question, the equation is

x^{2 }- 16x + 30 = 0

If 15 is a root, then it must satisfy the equation x^{2 }- 16x + 30 = 0,

But f(15) = 15^{2 }- 16(15) + 30 = 225 - 240 + 30 = 15

and so 15 should be subtracted from the equation.

So, the correct option is (c).

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

(a) x^{2 }- 9

(b) x^{2} + 9

(c) x^{2 } + 3

(d) x^{2 }- 3

If two zeros of the polynomial x^{3 } + x^{2 }- 9x - 9 are 3 and -3, then its third zero is

(a) -1

(b) 1

(c) -9

(d) 9

If x + 2 is a factor of x^{2 } + ax + 2b and a + b = 4, then

(a) a = 1, b = 3

(b) a = 3, b = 1

(c) a = -1, b = 5

(d) a = 5, b = -1

The polynomial which when divided by -x^{2 }+ x - 1 gives a quotient x - 2 and remainder is 3, is

(a) x^{3} - 3x^{2} + 3x - 5

(b) -x^{3} - 3x^{2} - 3x - 5

(c) -x^{3} + 3x^{2 }- 3x + 5

(d) x^{3} - 3x^{2} - 3x + 5

We know that

Dividend = Divisor × quotient + remainder

Then according to question,

Required polynomial

= (-x^{2} + x - 1) (x - 2) + 3

= -x^{3} + 2x^{2} + x^{2} -2x - x + 2 + 3

= -x^{3} + 3x^{2} - 3x + 5

So, the correct option is (c).

The number of polynomials having zeroes -2 and 5 is

a. 1

b. 2

c. 3

d. more than 3

Correct option: (d)

The polynomials 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.

If one of the zeroes of the quadratic polynomial (k
- 1)x^{2} + kx + 1 is - 3, then the value
of k is

a.

b.

c.

d.

The zeroes of the quadratic polynomial x^{2}
+ 99x + 127 are

a. Both positive

b. Both negative

c. both equal

d. One positive and one negative

The zeroes of the quadratic polynomial x^{2}
+ 99x + 127 are both negative since all terms are positive.

Hence, correct option is (b).

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

a. a = -7, b = -1

b. a = 5, b = -1

c. a = 2, b = -6

d. a = 0, b = -6

Given that one of the zeroes of the cubic polynomial
ax^{3} + bx^{2} + cx + d is zero, the product of the other
two zeroes is

a.

b.

c. 0

d.

The zeroes of the quadratic polynomial x^{2}
+ ax + a, a ≠ 0,

a. cannot both be positive

b. cannot both be negative

c. are always unequal

d. are always equal

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

a. b - a + 1

b. b - a - 1

c. a - b + 1

d. a - b - 1

## Chapter 2 - Polynomials Excercise 2.65

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

a.

b.

c.

d.

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

a. 10

b. -10

c. 5

d. -5

If the zeros of the quadratic polynomial ax^{2}
+ bx + c, c ≠ 0 are equal , then

a. c and a have opposite signs

b. c and b have opposite signs

c. c and a have the same sign

d. c and b have the same sign

It is given that the zeros of the quadratic
polynomial ax^{2} + bx + c, c ≠ 0 are
equal.

⇒ Discriminant = 0

⇒
b^{2} - 4ac = 0

⇒
b^{2} = 4ac

Now, b^{2} can never be negative,

Hence, 4ac also can never be negative.

⇒ a and c should have same sign.

Hence, correct option is (c).

If one of the zeros of a quadratic polynomial of the
form x^{2} + ax + b is the negative of the other, then it

a. has no linear term and constant term is negative.

b. has no linear term and the constant term is positive

c. can have a linear term but the constant term is negative

d. can have a linear term but the constant term is positive

Which of the following is not the graph of a quadratic polynomial?

The graph of a quadratic polynomial crosses X-axis at atmost two points.

Hence, correct option is (d).

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