RS Aggarwal Solutions for Class 10 Maths Chapter 2 Polynomials, contains solutions for all exercise 2C questions. RS Aggarwal solutions on exercise 2c is based on polynomials, types, zeros of a polynomial and relationship between zeros and coefficients of a quadratic Polynomial. Download the RS Aggarwal Solutions of Class 10 maths and sharpen your skills.

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Exercise 2A Solutions: 21 Questions (Short Answers)

Exercise 2B Solutions: 19 Questions (Short Answers)

**Exercise 2C Page No: 59**

**QUESTION 1: If one zero of the polynomial x^2 â€’ 4x + 1 is (2 + âˆš3), write the other zero.**

**SOLUTION: **

Given: (2 + âˆš3) is one of zero of polynomial x^2 â€’ 4x + 1.

To find: Other zero

Since given polynomial is a quadratic so it has only two zeros.

Let other zero be x.

Now,

Sum of zeros = -(coefficient of x)/(coefficient of x^2)

x + (2 + âˆš3) = -(-4/1) = 4

x + 2 + âˆš3 = 4

or x = 2 – âˆš3

Hence, the other zero is (2 -âˆš3).

**QUESTION 2: Find the zeros of the polynomial x^2 + x â€’ p (p + 1).**

**SOLUTION: **

Let f(x) = x^2 + x â€’ p (p + 1)

Above polynomial can be written as,

= x^2 + (p + 1) x â€“ px â€“ p (p + 1)

= x (x + (p + 1)) â€“ p (x + (p + 1))

= (x â€“ p)(x + (p + 1))

To find the zeroes of f(x), put f(x) = 0

(x â€“ p)(x + (p + 1)) = 0

either (x â€“ p) = 0 or (x + (p + 1)) = 0

x = p or x = – (p + 1)

Hence, the zeros of the given polynomial are p and â€“ (p + 1)

**QUESTION 3**

**Find the zeros of the polynomial x^2 â€“ 3x â€“ m (m + 3).**

**SOLUTION: **

Let f(x) = x^2 â€“ 3x â€“ m (m + 3)

Above polynomial can be written as,

f(x) = x^2 â€“ (m + 3)x + mx â€“ m(m + 3)

= x(x â€“ m â€“ 3) + m(x â€“ m â€“ 3)

= (x â€“ m â€“ 3)(x + m)

To find the zeroes of f(x), put f(x) = 0

(x â€“ m â€“ 3)(x + m) = 0

Either x â€“ m â€“ 3 = 0 or x + m = 0

x = m + 3 or x = -m

Required Zeros are (m + 3), -m

**QUESTION 4**

**Find Î±, Î² are the zeros of a polynomial such that Î± + Î²= 6 and Î±Î² = 4 then write the polynomial.**

**SOLUTION: **

Given : Î± + Î²= 6 = sum of zeros and

Î±Î² = 4 = Product of zeroes

We know that, if Î± and Î² are the zeros of the polynomial then the quadratic polynomial can be x^2 â€“ (Î± + Î²) x + Î±Î²

Now substituting the values, we get

x^2 â€“ 6x + 4, which is required polynomial.

**QUESTION 5**

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

**SOLUTION: **

Given: 2 is one of zeroes of the polynomial kx^2 + 3x + k , which means x = 2 will satisfy it.

k(2)^2 + 3( 2) + k = 0

4k + 6 + k= 0

5k + 6 = 0

k = -6 / 5

The value of k is -6 / 5

**QUESTION 6**

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

**SOLUTION: **

Given: 3 is one of zeroes of the polynomial 2x^2 + x + k , which means x = 3 will satisfy it.

2x^2 + x + k = 0

2(3)^2 + 3 + k = 0

18 + 3+ k = 0

k = -21

the value of k is -21

**QUESTION 7**

**If â€’4 is a zero of the polynomial x^2 â€’ x- (2k + 2) then find the value of k.**

**SOLUTION: **

Given: -4 is one of zeroes of the polynomial x^2 â€’ x- (2k + 2), which means x = -4 will satisfy it.

x^2 â€“ x â€“ (2k + 2) = 0

(-4)^2 â€“ (-4) â€“ 2k â€“ 2 = 0

16 + 4 â€“ 2k â€“ 2 = 0

-2k + 18 = 0

k = 9

The value of k is 9.

**QUESTION 8**

**If 1 is a zero of the polynomial ax^2 â€’ 3 (a â€’ 1) x â€’ 1 then find the value of a.**

**SOLUTION: **

Given: 1 is one of zeroes of the polynomial ax^2 â€’ 3 (a â€’ 1) x â€’ 1 , which means x = 1 will satisfy it.

a(1)^2 â€“ 3(a â€“ 1) x 1 â€“ 1 = 0

a â€“ 3a + 3 â€“ 1 = 0

-2a + 2 = 0

a = 1

The value of a is 1.

**QUESTION 9**

**If â€’2 is a zero of the polynomial 3x^2 + 4x + 2k then find the value of k.**

**SOLUTION: **

Given: -2 is one of zeroes of the polynomial 3x^2 + 4x + 2k** , **which means x = -2 will satisfy it.

3(-2)^2 + 4(-2) + 2k = 0

12 â€“ 8 + 2k = 0

4 + 2k=0

k = -2

The value of k is -2.

**QUESTION 10**

**Write the zeros of the polynomial x^2 â€’ x â€’ 6.**

**SOLUTION: **

Let f(x) = x^2 â€“ x â€“ 6

= x^2 â€“ 3x + 2x â€“ 6

= x(x â€“ 3) + 2(x â€“ 3)

= (x â€“ 3)(x + 2)\

To find the zeros of f(x), let sat f(x) = 0, we get

(x â€“ 3)(x + 2) = 0

Either x â€“ 3 = 0 or x + 2 = 0

x = 3 or x = -2

Therefore, 3, -2 are zeros.

**QUESTION 11**

**If the sum of the zeros of the quadratic polynomial kx^2 â€’ 3x + 5 is 1, write the value of k.**

**SOLUTION: **

Given: Sum of zeros of polynomial is kx^2 â€“ 3x + 5 is 1.

We know that,

Sum of zeros = -(coefficient of x)/(coefficient of x^2) = -(-3) / k = 3/k

From above results, we get

3/k = 1

Or k = 3

**QUESTION 12**

**If the product of the zeros of the quadratic polynomial x^2 â€’ 4x + k is 3 then write the value of k.**

**SOLUTION: **

Given: Product of zeros of polynomial is x^2 â€’ 4x + k is 3.

We know that,

Product of zeros = (constant term)/(coefficient of x^2) = k/1 = k

From above results, we get

K = 3

**QUESTION 13**

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

**SOLUTION: **

Given: (x + a) is a factor of 2x^2 + (2a + 5) x + 10.

Which shows that one of the zeros of the given polynomial is x + a = 0, i.e. â€“a.

Now, x = -a satisfy the given polynomial.

2 (-a)^2 + (2a + 5)(-a) + 10 = 0

2a^2 â€“ 2a^2 â€“ 5a + 10 = 0

5a = 10

a = 2

**QUESTION 14**

**If (a â€’ b), a and (a + b) are zeros of the polynomial 2x^3 â€’ 6x^2 + 5x â€“ 7, write the value of a.**

**SOLUTION: **

(a â€“ b), a, (a + b) are the zeros of 2x^3 â€“ 6x^2 + 5x â€“ 7

Sum of zeros = -(coefficient of x)/(coefficient of x^2)

a â€“ b + a + a + b = -(-6) / 2 = 3

3a = 3

or a = 1

**QUESTION 15**

**If x^3 + x^2 â€’ ax + b is divisible by (x^2 â€’ x), write the values of a and b.**

**SOLUTION: **

On dividing x^3 + x^2 â€’ ax + b by (x^2 â€’ x):

Remainder = (2 â€“ a) x + b

Since given polynomial is divisible by x^2 – x (given)

So remainder should be zero.

Therefore,

(2 â€“ a) x + b = 0

pnly possible if, 2 â€“ a = 0 and b = 0

a = 2 and b = 0

Hence, the values are : a = 2, b = 0

**QUESTION 16**

**If Î± and Î² are the zeros of the polynomial 2x^2 + 7x + 5, write the value of Î± + Î² + Î±Î².**

**SOLUTION: **

Given: Î± and Î² are the zeros of polynomial 2x^2 + 7x + 5

Î± + Î² = Sum of zeros = -(coefficient of x)/(coefficient of x^2) = -7/2

Î±Î² = Product of zeros = (constant term)/(coefficient of x^2) = 5/2

Î± + Î² + Î±Î² = (Î± + Î²) + Î±Î² = -7/2 + 5/2 = -1

**QUESTION 17**

**State division algorithm for polynomials.**

**SOLUTION: **

Division algorithm for polynomials states that: f(x) = q(x) g(x) + r(x)

Where f(x) and g(x) are any two polynomials with g(x) â‰ 0.

After dividing f(x) by g(x), we get two another polynomials, q(x) and r(x)

where r (x) = 0 and degree of r(x) < degree of g(x)

**QUESTION 18**

**The sum of the zeros and the product of zeros of a quadratic polynomial are -1/2 and â€’3 respectively. Write the polynomial.**

**SOLUTION: **

Sum of zeros = -1/2

Product of zeros = -3

We know that, general form of a Polynomial:

x^2 â€“ (Sum of zeros) x + product of zeros

This implies: x^2 â€“ (-1/2) + (-3)

x^2 + 1/2x – 3

**QUESTION 19**

**Write the zeros of the quadratic polynomial f(x) = 6x^2 â€’ 3.**

**SOLUTION: **

To find the zeros of the quadratic polynomial, put f(x) = 0

6x^2 â€“ 3 = 0

3 (2x^2 â€“ 1) = 0

2x^2 = 1

x^2 = 1/2

or x = Â±1/âˆš2

Zeros of the given polynomial are 1/âˆš2 and -1/âˆš2.

**QUESTION 20**

**Write the zeros of the quadratic polynomial f(x) = 4âˆš3x^2 + 5x â€’ 2âˆš3.**

**SOLUTION: **

f(x) = 4âˆš3x^2 + 5x â€’ 2âˆš3

To find the zeros of the quadratic polynomial, put f(x) = 0

4âˆš3x^2 + 5x â€’ 2âˆš3 = 0

4âˆš3 x^2 + 8x â€“ 3x – 2âˆš3 = 0

4x (âˆš3 x + 2) – âˆš3 (âˆš3 x + 2) = 0

(4x – âˆš3) (âˆš3 x + 2) = 0

Either (âˆš3 x + 2) = 0 or (4x – âˆš3) = 0

x = -2/âˆš3 or x = âˆš3/4

**QUESTION 21**

**If Î± and Î² are the zeros of the polynomial f(x) = x^2 â€’ 5x + k such that Î± â€’ Î² = 1, find the value of k.**

**SOLUTION: **

Given: Î± and Î² are zeroes of f(x) = x^2 â€’ 5x + k

and Î± – Î² = 1

We know that,

Î± + Î² = Sum of zeros = -(coefficient of x)/(coefficient of x^2) = 5

Î± Î² = Product of zeros = (constant term)/(coefficient of x^2) = k

Now solving Î± – Î² = 1 and Î± + Î² = 5, we get:

Î± = 3 and Î² = 2

Again: Î± Î² = k (putting value of Î± and Î²)

k = 6.

**QUESTION 22**

**If Î± and Î² are the zeros of the polynomial f(x) = 6x^2 + x â€’ 2, find the value of (Î±/ Î² + Î² / Î±)**

**SOLUTION: **

Given: Î± and Î² are zeroes of f(x) = 6x^2 + x – 2

To find: (Î±/ Î² + Î² / Î±)

Î± + Î² = Sum of zeros = -(coefficient of x)/(coefficient of x^2) = -1/6

Î± Î² = Product of zeros = (constant term)/(coefficient of x^2) = -1/3

Now,

(Î±/ Î² + Î² / Î±) = {(Î± + Î²)^2 – 2Î±Î²) / Î±Î² }

= (1/36 + 2/3) / (-1/3)

= -25/12

**QUESTION 23**

**If Î± and Î² are the zeros of the polynomial f(x) = 5x^2 â€’ 7x + 1, find the value of (1/Î± + 1/Î²)**

**SOLUTION: **

Given: Î± and Î² are zeroes of f(x) = 5x^2 â€’ 7x + 1

To find: (1/Î± + 1/Î²)

Î± + Î² = Sum of zeros = -(coefficient of x)/(coefficient of x^2) = 7/5

Î± Î² = Product of zeros = (constant term)/(coefficient of x^2) = 1/5

Now,

(1/Î± + 1/Î²) = (Î± + Î²) / Î±Î²

= 7/5 x 5/1

= 7

**QUESTION 24**

**If Î± and Î² are the zeros of the polynomial f(x) = x^2 + x â€“ 2, find the value of (1/Î± – 1/Î²)**

**SOLUTION: **

Given: Î± and Î² are zeroes of f(x) = x^2 + x â€“ 2

To find: (1/Î± – 1/Î²)

Î± + Î² = Sum of zeros = -(coefficient of x)/(coefficient of x^2) = -1

Î± Î² = Product of zeros = (constant term)/(coefficient of x^2) = -2

Now,

(1/Î± – 1/Î²) = (Î² – Î±)^2) / Î±Î²

= (Î² + Î±)^2 – 4Î±Î²) / (Î±Î²)^2

= 9/ 4

**QUESTION 25**

**If the zeros of the polynomial f(x) = x^3 â€“ 3x^2 + x + 1 are (a â€“ b), a and (a + b), find a and b.**

**SOLUTION: **

Given: Zeros of the polynomial x^3 â€“ 3x^2 + x + 1 are (a â€“ b), a and (a + b).

Now by using the relationship between the zeros of the quadratic polynomial we have:

Sum of zeros = -(coefficient of x)/(coefficient of x^2) = -1

a â€“ b + a + a + b = -1

3a = 3

a = 1

Product of zeros = (constant term)/(coefficient of x^2) = -1

(a â€“ b) (a) (a + b) = -1

(1 â€“ b) (1) (1 + b) = – 1

1 â€“ b^2 = – 1

b = Â±âˆš2

## RS Aggarwal Solutions for Class 10 Maths Chapter 2 Polynomials Exercise 2C

Class 10 Maths Chapter 2 Polynomials Exercise 2C is based on the following topics and subtopics:

- Value of a Polynomials at a given point
- Zeros of a Polynomials
- Relation between the zeros and coefficients of a quadratic Polynomial