# How do you use the remainder theorem to see if the n+5 is a factor of n^5-25n^3-7n^2-37n-18?

Oct 30, 2016

$\left(n + 5\right)$ is not a factor

#### Explanation:

Let $f \left(n\right) = {n}^{5} - 25 {n}^{3} - 7 {n}^{2} - 37 n - 18$
To see if $\left(n + 5\right)$ is a factor, we calculate $f \left(- 5\right)$
So $f \left(- 5\right) = {\left(- 5\right)}^{5} - 25 \cdot {\left(- 5\right)}^{3} - 7 {\left(- 5\right)}^{2} - 37 \cdot \left(- 5\right) - 18$
$= - 3125 + 3125 - 175 + 185 - 18 = - 8$
$f \left(- 5\right) \ne 0$
There is a remainder of $- 8$

Therefore $\left(n + 5\right)$ is not a factor

Oct 30, 2016

Analyse the value of the polynomial for $n = - 5$ to find $\left(n + 5\right)$ is not a factor

#### Explanation:

Given:

$f \left(n\right) = {n}^{5} - 25 {n}^{3} - 7 {n}^{2} - 37 n - 18$

The remainder theorem tells us that $\left(n + 5\right)$ is a factor of $f \left(n\right)$ if and only if $f \left(- 5\right) = 0$

Observe that all of the terms of $f \left(n\right)$ except the constant term are divisible by $n$:

$f \left(n\right) = {n}^{5} - 25 {n}^{3} - 7 {n}^{2} - 37 n - 18$

$\textcolor{w h i t e}{f \left(n\right)} = n \left({n}^{4} - 25 {n}^{2} - 7 n - 37\right) - 18$

So if $n$ is not a factor of the constant term then $f \left(n\right) \ne 0$...

$f \left(\textcolor{b l u e}{- 5}\right) = \left(\textcolor{b l u e}{- 5}\right) \left({\left(\textcolor{b l u e}{- 5}\right)}^{4} - 25 {\left(\textcolor{b l u e}{- 5}\right)}^{2} - 7 \left(\textcolor{b l u e}{- 5}\right) - 37\right) - 18$

$\textcolor{w h i t e}{f \left(\textcolor{w h i t e}{- 5}\right)} = \left(\textcolor{b l u e}{- 5}\right) \left({\left(\textcolor{b l u e}{- 5}\right)}^{4} - 25 {\left(\textcolor{b l u e}{- 5}\right)}^{2} - 7 \left(\textcolor{b l u e}{- 5}\right) - 37\right) - 5 \cdot 4 + 2$

$\textcolor{w h i t e}{f \left(\textcolor{w h i t e}{- 5}\right)} = 5 k + 2 \text{ }$ for some integer $k$

$\textcolor{w h i t e}{f \left(\textcolor{w h i t e}{- 5}\right)} \ne 0$

Since $f \left(- 5\right) \ne 0$ we can deduce that $\left(n + 5\right)$ is not a factor.