## Chapter 2 Polynomials R.D. Sharma Solutions for Class 10th Math Exercise 2.3

**Exercise 2.3**

(i) f(x) = x

^{3}- 6x

^{2}+ 11x - 6, g(x) = x

^{2}+ x + 1

(ii) f(x) 10x

^{4}+ 17x

^{3}- 62x

^{2}+ 30x - 105(x) = 2x

^{2}+ 7x + 1

(iii) f(x) = 4x

^{3}+ 8x

^{2}+ 8x + 7:9(x) = 2x

^{2}- x + 1

(iv) f(x) = 15x

^{3}- 20x

^{2}+ 13x - 12; g(x) = x

^{2}- 2x + 2

**Solution**

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

(i) g(t) = t

^{2}- 3; f(t) = 2t

^{4}+ 3t

^{3}- 2t

^{2}- 9t

(ii) g(x) = x

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

^{5}- 4x

^{3}+ x

^{2}+ 3x + 1

(iii) g(x) = 2x

^{2}- x + 3, f(x) = 6x

^{5}- x

^{4}+ 4x

^{3}- 5x

^{2}- x - 15

**Solution**

3. Obtain all zeros of the polynomial f(x) = 2x

^{4}+ x

^{3}- 14x

^{2}- 19x - 6, if two of its zeors are -2 and -1.

**Solution**

4. Obtain all zeros of f(x) = x

^{3}+ 13x

^{2}+ 32x + 20, if one of its zeros is -2.

**Solution**

5. Obtain all zeros of the polynomial f(x) = x

^{4}- 3x

^{2}= x

^{2}+ 9x - 6 if two of its zeros are -√3, and √3 .

**Solution**

**Solution**

7. Find all the zeroes of the polynomial x

^{4}+ x

^{3}- 34x

^{2}- 4x + 120, if two of its zeroes are 2 and -2 .

**Solution**

8. Find all zeros of the polynomial 2x

^{4}+ 7x

^{3}- 19x

^{2}- 14x + 30, if two of its zeros are √2 and -√2 .

**Solution**

9. Find all the zeros of the polynomial 2x

^{3}+ x

^{2}- 6x - 3, if two of its zeros are -√3 and √3.

**Solution**

10. Find all the zeros of the polynomial x

^{3}+ 3x

^{2}- 2x - 6, if two of its zeros are -√2 and √2.

**Solution**

11. What must be added to the polynomial f(x) = x

^{4}+ 2x

^{3}- 2x

^{2}+ x -1 so that the resulting polynomial is exactly by x

^{2}+ 2x - 3 ?

**Solution**

12. What must be subtracted from the polynomial f(x) = x

^{4}+ 2x

^{3}- 13x

^{2}- 12x + 21 so that the resulting polynomial is exactly divisible by x

^{2}- 4x + 3 ?

**Solution**

13. Given that √2 is a zero of the cumbic polynomial 6x

^{3}+ √2x

^{2}– 10x - 4√2 , find its other two zeroes.

**Solution**

We know that if x = a is a zero of a polynomial, and then x -1 is a factor of f (x).

It is given that √2 is a zero of the cubic polynomial f (x) = 6x

^{3}+ √2x

^{2}– 10x - 4√2 .

Therefore, x - √2 is a factor of f(x) . Now, we divide 6x

^{3}+ √2x

^{2}– 10x - 4√2 by x - √2 to find the other zeroes of f(x).

∴ Quotient = 6x

^{2}+ 7√2x + 4 and remainder = 0 .

By using division algorithm , we have f(x) = g(x) × q(x) + r(x) .

f(x) = (x - √2) (6x

^{2}+ 7√2x + 4) + 0

= (x - √2) (√2x + 1) (3√2x + 4)

Hence, the other two zeroes of the given polynomial are – 1/√2 and – 4/3√2 .

14 . Given that x - √5 is a factor of the cubic polynomial x

^{3}– 3√5x

^{2}+ 13x - 3√5 , find all the zeroes of the polynomial .

**Solution**

We know that if x = a is a zero of a polynomial , and then x –a is a factor of f(x) .

It is given that x - √5 is a factor of f(x) = x

^{3}– 3√5x

^{2}+ 13x - 3√5.

Now, we divide x

^{3}– 3√5x

^{2}+ 13x - 3√5 by x -√5 to find the other zeroes of f(x) .

∴ Quotient = x

^{2}– 2√5x + 3 and remainder = 0 .

By using division algorithm , we have f(x) = g(x) × q(x) + r(x) .

f(x) = (x- √5 ) (x

^{2}- 2√5x + 3) + 0

= (x- √5) [x-(√5 + √2)] [x-(√5 - √2)]

Hence, the zeroes of the given polynomial are √5 , √5 + √2 and √5 - √2 .