A question about remainder theorem

Question

$\displaystyle \frac{n^4 + 10n^3 + 21n^2 + 6n − 8}{n + 2}$

Prove how the binomial is a factor of the polynomial. I keep getting a remainder. How am I doing it wrong?

Perhaps you are wrongly substituting $n=2$ instead of $n=-2$. — user11977, Mar 20 '14 at 23:40
It is hard to tell what you are doing wrong if you don't post what you've done. — robjohn, Mar 20 '14 at 23:41
When dividing a polynomial $p(x)$ by a linear factor, $(x-\alpha)$, the remainder is $p(\alpha)$. This is since we can write $p(x) = q(x)\cdot (x-\alpha) + r(x)$ where $r$ has degree strictly less than $(x-\alpha)$ (i.e., it's a constant term). In this case, evaluate the numerator at n=-2 to find the remainder. — ah11950, Mar 20 '14 at 23:42

score 2 · Answer 1 · answered Mar 20 '14 at 23:43

2

Hint: Try multiplying: $(n+2)(n^3+8n^2+5n-4)$

Double check your division.

answered Mar 20 '14 at 23:43

robjohn

score 2 · Answer 2 · answered Mar 20 '14 at 23:47

2

If all you need is to show that $n+2$ is a factor of the polynomial, you could just show that $P(-2) = 0$.

answered Mar 20 '14 at 23:47

MT_

2 Answers2