Question

Find the quotient and remainder when `x^3+3px+q` is divided by `(x-a)^2`?

A. Quotient is `x` and remainder is `a^3+3pa+q`
B. Quotient is `x^2+3p` and remainder is `a^3+3pa+q`
C. Quotient is `x+2` and remainder is `3p-a^3+q`
D. Quotient is `x+2` and remainder is `3(p+a^2)x-2a^3+q`

Answer 1

Answer is D.

Explanation:

The highest degree of `x^3+3px+q` is `x^3`

and highest degree of `(x-a)^2=x^2-2ax+a^2` is `x^2`

hence quotient will have `x^3/x^2` or `x` as the highest degree

`x^3+3px+q=x(x^2-2ax+a^2)+2ax^2-a^2x+3px+q`

`=x(x^2-2ax+a^2)+2a(x^2-2ax+a^2)+4a^2x-2a^3-a^2x+3px+q`

`=(x+2a)(x^2-2ax+a^2)+3(p+a^2)x-2a^3+q`

Hence, quotient is `x+2` and remainder is `3(p+a^2)x-2a^3+q`

and answer is D.