For which value of a and …

We will find zeroes of p(x)  which are not the zeroes of q(x) by using Factor theorem and Euclid’s division algorithm.

By factor theorem if q(x) is a factor of p(x), then r(x) must be zero.

p(x) = x⁵ – x⁴ – 4x³ + 3x² + 3x + b

q(x) = x³ + 2x² + a

So, by factor theorem remainder must be zero i.e.,

r(x) = 0
Value of r(x) is - ax² - x² + 3ax + 3x - 2a + b .
- ax² - x² + 3ax + 3x - 2a + b = 0x² + 0x + 0
⇒ -(a + 1)x² + (3a + 3)x + (b – 2a) = 0x² + 0x + 0

Comparing the coefficients of x², x and constant. on both sides, we get

-(a + 1) = 0 and 3a + 3 = 0 and b – 2a = 0
a + 1 = 0
a = -1
and  3a + 3 = 0
3a = - 3
a = -1
Put a = -1 in b - 2a = 0
Then b – 2(-1) = 0
⇒ b + 2 = 0
⇒ b = -2

For a = -1 and b = -2, zeroes of q(x) will be zeroes of p(x).

For zeroes of p(x),  p(x) = 0

⇒ (x³ + 2x² + a)(x² – 3x + 2) = 0 [∵ a = -1]

⇒ [x³ + 2x² – 1][x² – 2x – 1x + 2] =0

⇒ (x³ + 2x² – 1)[x(x – 2) – 1(x – 2) = 0

⇒ (x³ + 2x² – 1) (x – 2) (x – 1) = 0
⇒   (x – 2)  = 0 and (x – 1) = 0
⇒  x = 2 and x = 1

Hence, x = 2 and 1 are not the zeroes of q(x).