RD SHARMA SOLUTION CHAPTER- 2 Polynomials| CLASS 10TH MATHEMATICS-EDUGROWN

Chapter 2 Polynomials Exercise Ex. 2.1

Question 1(i)

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

f(x) = x² – 2x – 8 Solution 1(i)

x² – 2x – 8 = x² – 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) = x² – 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.Question 1(ii)

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

g(s) = 4s² – 4s + 1 Solution 1(ii)

4s² – 4s + 1 = 4s² – 2s – 2s + 1 = 2s(2s – 1) – (2s – 1) = (2s – 1)(2s – 1) The zeroes of the quadratic equation are and . Let ∝ =  and β =  Consider4s² – 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.Question 1(iii)

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

h (t) = t² – 15 Solution 1(iii)

h (t) = t² – 15 = (t + √15)(t – √15)  The zeroes of the quadratic equation areand . Let ∝ =  and β =  Considert² – 15 = t² – 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.Question 1(iv)

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

f(s) = 6x² – 3 – 7x Solution 1(iv)

f(s) = 0 6x² – 3 – 7x =0 6x² – 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 =  Consider6x² – 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.Question 1(v)

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

Solution 1(v)

 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.Question 1(vi)

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

Solution 1(vi)

 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.Question 1(vii)

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

Solution 1(vii)

 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.Question 1(viii)

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

Solution 1(viii)

 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.Question 1(ix)

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

Solution 1(ix)

 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.Question 1(x)

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

Solution 1(x)

 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.Question 1(xi)

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

Solution 1(xi)

 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.Question 1(xii)

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

Solution 1(xii)

 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.Question 2(i)

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.

Solution 2(i)

Question 2(ii)

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.

Solution 2(ii)

Question 2(iii)

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.

Solution 2(iii)

Question 2(iv)

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.

Solution 2(iv)

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

(i)



(ii)



(iii)



(iv)



(v)



(vi)



(vii)



(viii)

 

Chapter 2 Polynomials Exercise Ex. 2.2

Question 1Verify 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:

Solution 1

On comparing the given polynomial with the polynomial ax³ + bx² + 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.

Question 2

Solution 2

Question 3

Find all zeroes of the polynomial 3x³ + 10x² – 9x – 4, if one of its zeroes is 1.Solution 3

Let f(x) = 3x³ + 10x² – 9x – 4 

As 1 is one of the zeroes of the polynomial, so (x – 1) becomes the factor of f(x).

Dividing f(x) by (x – 1), we have

Hence, the zeroes are  Question 4

If 4 is a zero of the cubic polynomial x³ – 3x² – 10x + 24, find its other two zeroes.Solution 4

Let f(x) = x³ – 3x² – 10x + 24 

As 4 is one of the zeroes of the polynomial, so (x – 4) becomes the factor of f(x).

Dividing f(x) by (x – 4), we have

Hence, the zeroes are 4, -3 and 2.Question 5

Solution 5

Question 6

Solution 6

Chapter 2 Polynomials Exercise Ex. 2.3

Question 1 (i)

Solution 1 (i)

Question 1 (ii)

Solution 1 (ii)

Question 1 (iii)

Solution 1 (iii)

Question 1 (iv)

Solution 1 (iv)

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Find all zeros of the polynomial 2x⁴ – 9x³ + 5x² + 3x – 1, if two of its zeros are  Solution 11

Let f(x) = 2x⁴ – 9x³ + 5x² + 3x – 1 

As  are two of the zeroes of the polynomial, so  becomes the factor of f(x).

Dividing f(x) by  we have

Hence, the other zeros  Question 12

For what value of k, is the polynomial f(x) = 3x⁴ – 9x³ + x² + 15x + k completely divisible by 3x² – 5?Solution 12

Let f(x) = 3x⁴ – 9x³ + x² + 15x + k 

As f(x) is completely divisible by 3x² – 5, it becomes one of the factors of f(x).

Dividing f(x) by 3x² – 5, we have

As (3x² – 5) is one of the factors, the remainder will be 0.

Therefore, k + 10 = 0

Thus, k = -10.Question 13

Solution 13

Question 14

Solution 14

Chapter 2 Polynomials Exercise 2.61

Question 1

(a) 1

(b) -1

(c) 0

(d) None of theseSolution 1

Question 2

Solution 2

So, the correct option is (d).Question 3

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

(a) 2

(b) -2

(c) 1

(d) -1Solution 3

Chapter 2 Polynomials Exercise 2.62

Question 4

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

(a) 2

(b) 4

(c) -2

(d) -4Solution 4

So, the correct option is (b).Question 5

(a) x² + qx + p

(b) x² – px + q

(c) qx² + px + 1

(d) px² + qx + 1Solution 5

Question 6

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

(a) c – 1

(b) 1 – c

(c) c

(d) 1 + cSolution 6

Question 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 =

(a) 1

(b) 0

(c) -1

(d) 2Solution 7

Given that (α + 1) (β + 1) = 0

So, the correct option is (a).Question 8

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

(a) c = 0

(b) c > 0

(c) c < 0

(d) None of theseSolution 8

We know that, if the quadratic equation ax² + 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).Question 9

If the diagram in figure show the graph of the polynomial f(x) = ax² + 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 < 0Solution 9

Question 10

Figure shows the graph of the polynomial f(x) = ax² + 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 < 0Solution 10

Question 11

Solution 11

Question 12

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

(a) 2p³ = pq – r

(b) 2p³ = pq + r

(c) p³ = pq – r

(d) None of theseSolution 12

Chapter 2 Polynomials Exercise 2.63

Question 13

Solution 13

So, the correct option is (b).Question 14

Solution 14

Question 15

In Q. No. 14, c =

(a) b

(b) 2b

(c) 2b²

(d) -2bSolution 15

So, the correct option is (c).Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

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

Solution 21

Question 22

Solution 22

Question 23

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

(a) -4

(b) 4

(c) 6

(d) -6Solution 23

Chapter 2 Polynomials Exercise 2.64

Question 24

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

(a) 1

(b) 2

(c) 4

(d) 5Solution 24

We know that, if α and β are roots of ax² + bx + c = 0 then they must satisfy the equation.

According to the question, the equation is

x² – 5x + 4 = 0

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

x² – 5x + 4 = 0

but f(3) = 3² – 5(3) + 4 = -2

so, 2 has to be added in the equation.

So, the correct option is (b).Question 25

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

(a) 30

(b) 14

(c) 15

(d) 16Solution 25

We know that, if α and β are roots of ax²  + bx + c = 0, then α and β must satisfy the equation.

According to the question, the equation is

x² – 16x + 30 = 0

If 15 is a root, then it must satisfy the equation x² – 16x + 30 = 0, 

But f(15) = 15² – 16(15) + 30 = 225 – 240 + 30 = 15

and so 15 should be subtracted from the equation.

So, the correct option is (c).Question 26

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

(a) x² – 9

(b) x² + 9

(c) x²  + 3

(d) x² – 3Solution 26

Question 27

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

(a) -1

(b) 1

(c) -9

(d) 9Solution 27

Question 28

Solution 28

Question 29

If x + 2 is a factor of x²  + 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 = -1Solution 29

Question 30

The polynomial which when divided by -x² + x – 1 gives a quotient x – 2 and remainder is 3, is

(a) x³ – 3x² + 3x – 5

(b) -x³ – 3x² – 3x – 5

(c) -x³ + 3x²  – 3x + 5

(d) x³ – 3x² – 3x + 5Solution 30

We know that 

Dividend = Divisor × quotient  + remainder

Then according to question,

Required polynomial

= (-x² + x – 1) (x – 2) + 3

= -x³ + 2x² + x² -2x – x + 2 + 3

= -x³ + 3x² – 3x + 5

So, the correct option is (c).Question 31

The number of polynomials having zeroes -2 and 5 is

a. 1

b. 2

c. 3

d. more than 3Solution 31

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.Question 32

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

a. 

b. 

c. 

d. Solution 32

Question 33

The zeroes of the quadratic polynomial x² + 99x + 127 are

a. Both positive

b. Both negative

c. both equal

d. One positive and one negativeSolution 33

The zeroes of the quadratic polynomial x² + 99x + 127 are both negative since all terms are positive.

Hence, correct option is (b).Question 34

If the zeroes of the quadratic polynomial x² + (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 = -6Solution 34

Question 35

Given that one of the zeroes of the cubic polynomial ax³ + bx² + cx + d is zero, the product of the other two zeroes is

a. 

b. 

c. 0

d. Solution 35

Question 36

The zeroes of the quadratic polynomial x² + ax + a, a ≠ 0,

a. cannot both be positive

b. cannot both be negative

c. are always unequal

d. are always equalSolution 36

Question 37

If one of the zeros of the cubic polynomial x³ + ax² + 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 – 1Solution 37

Chapter 2 Polynomials Exercise 2.65

Question 38

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

a. 

b. 

c. 

d. Solution 38

Question 39

If one zero of the quadratic polynomial x² + 3x + k is 2, then the value of k is

a. 10

b. -10

c. 5

d. -5Solution 39

Question 40

If the zeros of the quadratic polynomial ax² + 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 signSolution 40

It is given that the zeros of the quadratic polynomial ax² + bx + c, c ≠ 0 are equal.

⇒ Discriminant = 0

⇒ b² – 4ac = 0

⇒ b² = 4ac

Now, b² can never be negative,

Hence, 4ac also can never be negative.

⇒ a and c should have same sign.

Hence, correct option is (c).Question 41

If one of the zeros of a quadratic polynomial of the form x² + 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 positiveSolution 41

Question 42

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

Solution 42

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

Hence, correct option is (d).