Prove $3 \mid x-2 \implies 3 \mid (x^2 - x+1)$ using division algorithm

Question

I can't figure out how to prove the following implication using the division algorithm:

$$3 \mid x-2 \implies 3 \mid (x^2 - x+1)$$

It seems simple enough. Does anyone know how?

Answer 1

Hint: Let $A(x)=x^2-x+1$. Then $$A(x)=(x-2)Q(x)+R,$$ for some polynomial $Q$ with integer coefficients, and some constant $R$. Moreover, $R=A(2)=3$.

Answer 2

The essence of executing the division algorithm lies in $x^2 - x + 1 = (x-2)(x+1) + 3$.

That is, $\dfrac{x^2 - x + 1}{x-2} = x+1$ with a remainder of $3$.