Polynomial remainder and factor theorems

[Dean54321]

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]

(If P(x) is linear, then you can't divide by [MATH](x-1)(x+2)[/MATH] and get a remainder, so we'll assume they meant to say that P(x) is at least quadratic).

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

Now substitute [MATH]x=1[/MATH] and [MATH] x=-2[/MATH] into equation (1) and use the facts that you know: [MATH]P(1)=1[/MATH] and [MATH] P(-2)=-8[/MATH], to find [MATH]a[/MATH] and [MATH]b[/MATH].