**EXERCISE 2.4**

**EXERCISE 2.4**

**1.Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also verify the relationship between the zeroes and the coefficients in each case:**

### (i) 2x^{2}+ x^{2} – 5x + 2; _{}, 1,-2

### (ii) x^{3} – 4x^{2} + 5x – 2; 2, 1, 1

**Sol.** (i) Comparing the given polynomial with ax^{3}+ bx^{2} + cx + d, we get*a *= 2, *b*= 1, *c *= – 5 and* d* = 2.

*(ii)* Comparing the given polynomial with ax^{3}+ bx^{2} + cx + d, we get

a = 1, b = – 4, c = 5 and d = – 2.

p(2) = (2)^{3}– 4(2)^{2}+ 5(2) – 2 = 8 – 16 + 10 – 2 = 0

p(1) = (1)^{3} – 4(1)^{2} + 5(1) -2=1-4+5-2 = 0

=> 2, 1 and 1 are the zeroes of x^{3} – 4x^{2} + 5x – 2.

So,* a* = 2, = 1 and* y *= 1.

**2.Find a cubic polynomial with the sum of the product of its zeroes taken two at a time, and the product of its zeroes as 2, – 7, – 14 respectively.**

Sol. Let the cubic polynomial be ax^{3}+ bx^{2} + cx + d, and its zeroes be a, _{} and y.

Then,

If a = 1, then b = – 2, c = – 7 and d = 14.

So, one cubic polynomial which fits the given conditions will be x^{3} – 2x^{2} – 7x + 14.

**3.If the zeroes of the polynomial x**^{3}-3x^{2}+ x + 1 are *a – b, a, a + b*, find *a *and *b.*

**Sol.** Since (a – b), a, (a + b) are the zeroes of the polynomial x^{3}-3x^{2}+ x + 1

^{3}-3x

^{2}+ x + 1 are

*a – b, a, a + b*, find

*a*and

*b.*

p(x) = x^{4} – 6x^{3}– 26x^{2} + 138x – 35

= (x^{2} – 4x + 1) (x^{2} – 2x – 35)

= (x^{2}-4x+ 1)(x^{2} – 78 + 58 – 35)

= (x^{2} – 4x + 1) (x (x – 7) + 5(x – 7)]

= (x^{2}– 4x + 1)(x + 5)(x- 7)

(x + 5) and (x – 7) are other factors of p(x)

=> 5 and 7 are other zeroes of the given polynomial.

**5.If the polynomial x**^{4}– 6x^{3} + 16x^{2} – 25x + 10 is divided by another polynomial *x*^{2} – 2x + k, the remainder comes out to be* x + a*, find k and *a.*

^{4}– 6x

^{3}+ 16x

^{2}– 25x + 10 is divided by another polynomial

*x*, the remainder comes out to be

^{2}– 2x + k*x + a*, find k and

*a.*

**Sol.** Let us divide x^{4}– 6x^{3} + 16x^{2} – 25x + 10 by *x ^{2} – 2x + k*

22 – 2x + k.