Question

pls give the full sol. to the question

Answer 1

ax³ + bx² + cx + d is exactly divisible by (x + 1) and (x + 2).
means, (x + 1) and (x + 2) are factors of polynomial ax³ + bx² + cx + d .

or, x = -1 and -2 are zeros of polynomial ax³ + bx² + cx + d

at x = -1
a(-1)³ + b(-1)³ + c(-1) + d = 0
-a + b - c + d = 0
c = -a + b + d .......(1)

at x = -2
a(-2)³ + b(-2)² + c(-2) + d = 0
-8a + 4b - 2c + d = 0
from equation (1),
-8a + 4b -2(-a + b + d) + d = 0
-8a + 4b + 2a - 2b - 2d + d = 0
-6a + 2b - 2d + d = 0
-6a + 2b - d = 0
6a - 2b + d = 0

hence, option (D) is correct.

Answer 2

In the attachments I have answered this problem.

I have applied remainder theorem to solve this problem.

Remainder theorem:

When p(x) is divided by (x-a) the remainder is p(a)

See the attachment for detailed solution