Without using long division find the remainder when f(x) is divided by g(x).

Question

Without using long division find the remainder when f(x) is divided by g(x).

f(x)= x^4 - 5x^3 + 6x^2 - 7
g(x)=(x-1)(x-3)

I don't really get how to solve it without using long division. I taught for using the reminder theorem g(x) has to be linear?

Answer 1

We have $f(x)=q(x)g(x)+l(x)$, where $l(x)$ has shape $ax+b$.

So $f(1)=a+b$ and $f(3)=3a+b$. Now we have two linear equations in two unknowns, and can solve for $a$ and $b$.