The reaminder theorem

Quandaries and Queries

Name: Ron
Who is asking: Parent
Level of the question: Secondary

Question: If f(x) is divided by (x-3), the remainder is 5, and if it is divided by (x+1), the remainder is -3. Find the remainder when f(x)is divided by (x-3)(x+1).

Hi Ron,

You cannot say anything unless f(x) is a polynomial. if so then the factor theorem says that
f(x) = (x-3) g(x) + 5 = (x+1) h(x) - 3 for some polynomials g(x) and h(x).

Now plug x = 3 into this equality to get a formula for h(3) (namely, h(3) = 2). The factor theorem then gives you a formula for h(x), namely h(x) = (x-3) k(x) + 2.

Just put this expression for h(x) into the second equation for f(x) and read off the answer.

Chris