# Polynomial remainder and factor theorems

#### Dean54321

##### New member
When P(x) is divided by x-1, the remainder is 1, and P(x) is divided by x+2, the remainder is -8. Find the remainder, if P(x) is divided by (x-1)(x+2).

Is there a rule for this? because we don't know the polynomial P(x).

#### lex

##### Full Member
(If P(x) is linear, then you can't divide by $$\displaystyle (x-1)(x+2)$$ and get a remainder, so we'll assume they meant to say that P(x) is at least quadratic).

$$\displaystyle P(x)=(x-1)(x+2)Q(x)+R(x) \quad$$ (1)
where R(x) is $$\displaystyle ax+b$$
(Note you are dividing by a quadratic, so the remainder may be linear).

Now substitute $$\displaystyle x=1$$ and $$\displaystyle x=-2$$ into equation (1) and use the facts that you know: $$\displaystyle P(1)=1$$ and $$\displaystyle P(-2)=-8$$, to find $$\displaystyle a$$ and $$\displaystyle b$$.