Find the remainder when the polynomial x^5 - 4x^3 + 7x^2 - 3 is divided by (x - 1)(x + 2). I think there is a way to do this using Remainder Theorem, but I have no idea how.
We can use long division
I got 4x+9 but I'm not sure
i suggest you search up polynomial division, there are some great tutorials like khan academy that are free
Remainder theorem will get you there.
\(\displaystyle \frac{x^{5}-4x^{3}+7x^{2}-3}{(x-1)(x+2)}=q(x)+\frac{r(x)}{(x-1)(x+2)}.\)
q(x) will be a cubic and r(x) will be linear, (one degree lower than the quadratic on the bottom line), so let r(x) = ax + b.
Multiply throughout by (x - 1)(x + 2) to get
\(\displaystyle x^{5}-4x^{3}+7x^{2}-3=q(x)(x-1)(x+2)+ax+b\),
and now substitute x = 1 and x = -2.
That gets you a pair of simultaneous equations in a and b which you can solve.
