Question

The remainder when x^3 + 3px + q is divided by (x-a)^2, is
The answer is 3 (p + a^2)x + q - 2a^3. Please explain it with steps.

Answer 1

${\huge{\red{\underline{Answer}}}}$

Let f(x)=x³+3px+q=(x-a)²Q(x)+A(x-a)+B

f’(x)=2(x-a)Q(x)+Q’(x)(x-a)²+A

f’(a)=A

f’(x)=3x²+3p

f’(a)=3a²+3p=A→3p=A-3a²

f(a)=B=a³+3ap+q

reqd remainder=A(x-a)+B=3(a²+p)(x-a)+a³+3ap+q