Factor and Remainder Theorem

Question

Please help me with this exam style question from the IGCSE 2020 Specimen Paper. The polynomial $$p(x)=2x^3 -3x^2 +qx+56$$ has $x-2$ as a factor. Show that $q=-30$.

Answer 1

If $x-2$ is a factor, then $p(2)= 2\cdot 2^3 -3 \cdot 2^2 + q\cdot 2 + 56 = 0$. $$\Rightarrow 2q = -60$$

Answer 2

You want $P(2)=0$

What is $ P(2)$ when $ p(x)=2x^3 -3x^2 +qx+56$?

Let $x=2$ and solve for $q$