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

**Exercise 2.3**

(i) f(x) =Â x

^{3}Â - 6x

^{2}Â + 11x - 6, g(x) =Â x

^{2}Â + x + 1

(ii) f(x) 10x

^{4}Â + 17x

^{3}Â - 62x

^{2}Â + 30x - 105(x) = 2x

^{2}Â + 7x + 1

(iii) f(x) = 4x

^{3}Â + 8x

^{2}Â + 8x + 7:9(x) = 2x

^{2}Â - x + 1

(iv) f(x) = 15x

^{3}Â - 20x

^{2}Â + 13x - 12; g(x) =Â x

^{2}Â - 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}Â - 3; f(t) = 2t

^{4}Â + 3t

^{3}Â - 2t

^{2}Â - 9t

(ii) g(x) = x

^{2}Â - 3x + 1, f(x) = x

^{5}Â - 4x

^{3}Â + x

^{2}Â + 3x + 1

(iii) g(x) = 2x

^{2}Â - x + 3, f(x) = 6x

^{5}Â - x

^{4}Â + 4x

^{3}Â - 5x

^{2}Â - x - 15

**Solution**

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

^{4}Â +Â x

^{3}Â - 14x

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

**Solution**

4. Obtain all zeros of f(x) =Â x

^{3}Â + 13x

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

**Solution**

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

^{4}Â - 3x

^{2}Â =Â x

^{2}Â + 9x - 6 if two of its zeros are -âˆš3, and âˆš3 .

**Solution**

**Solution**

7. Find all the zeroes of the polynomial x

^{4}Â +Â x

^{3}Â - 34x

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

**Solution**

8. Find all zeros of the polynomial 2x

^{4}Â + 7x

^{3}Â - 19x

^{2}Â - 14x + 30, if two of its zeros are âˆš2 and -âˆš2 .

**Solution**

9. Find all the zeros of the polynomial 2x

^{3}Â + x

^{2}Â - 6x - 3, if two of its zeros are -âˆš3 and âˆš3.

**Solution**

10. Find all the zeros of the polynomial x

^{3}Â + 3x

^{2}Â - 2x - 6, if two of its zeros are -âˆš2 and âˆš2.

**Solution**

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

^{4}Â + 2x

^{3}Â - 2x

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

^{2}Â + 2x - 3 ?

**Solution**

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

^{4}Â + 2x

^{3}Â - 13x

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

^{2}Â - 4x + 3 ?

**Solution**

13. Given that âˆš2 is a zero of the cumbic polynomial 6x

^{3}Â + âˆš2x

^{2}Â â€“ 10x - 4âˆš2 , 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 âˆš2 is a zero of the cubic polynomial f (x) = 6x

^{3}Â + âˆš2x

^{2}Â â€“ 10x - 4âˆš2 .

Therefore, x - âˆš2 is a factor of f(x) . Now, we divide 6x

^{3}Â + âˆš2x

^{2}Â â€“ 10x - 4âˆš2 by x - âˆš2 to find the other zeroes of f(x).

âˆ´ Quotient = 6x

^{2}Â + 7âˆš2x + 4 and remainder = 0 .

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

f(x) = (x - âˆš2) (6x

^{2}Â + 7âˆš2x + 4) + 0

= (x - âˆš2) (âˆš2x + 1) (3âˆš2x + 4)

Hence, the other two zeroes of the given polynomial are â€“ 1/âˆš2 and â€“ 4/3âˆš2 .

14 . Given that x - âˆš5 is a factor of the cubic polynomialÂ x

^{3}Â â€“ 3âˆš5x

^{2}Â + 13x - 3âˆš5 , find all the zeroes of the polynomial .

**Solution**

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

It is given that x - âˆš5 is a factor of f(x) =Â x

^{3}Â â€“ 3âˆš5x

^{2}Â + 13x - 3âˆš5.

Now, we divideÂ x

^{3}Â â€“ 3âˆš5x

^{2}Â + 13x - 3âˆš5Â by x -âˆš5 to find the other zeroes of f(x) .

âˆ´ Quotient =Â x

^{2}Â â€“ 2âˆš5x + 3 and remainder = 0 .

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

f(x) = (x- âˆš5 ) (x

^{2}Â - 2âˆš5x + 3) + 0

= (x- âˆš5) [x-(âˆš5 + âˆš2)] [x-(âˆš5 - âˆš2)]

Hence, the zeroes of the given polynomial are âˆš5 , âˆš5 + âˆš2 and âˆš5 - âˆš2 .