Question

If you divide $f\left(x\right)=x^3+3x^2-kx-12$ by $\left(x-3\right)$, you get remainder $30$. Find $k$ and also find the quotient.

Answer 1

$f\left(x\right)=x^3+3x^2-kx-12$$

According to remainder theorem

$f\left(3\right)=30$

$⟹(3{)}^2+3(3{)}^3-k\left(3\right)-12=30$

$⟹27+27-3k-12=30⟹3k=54-12-30=12$

$⟹k=4$

$f\left(x\right)=x^3+3x^2-4x-12$

$⟹\frac{x^3+3x^2-4x-12}{x-3}=x^2+6x+14$

So, Quotient $=x^2+6x+14$