# How do you use the remainder theorem and the factor theorem to determine whether (c+5) is a factor of (c^4+7c^3+6c^2-18c+10)?

Oct 4, 2015

Evaluate $\left({c}^{4} + 7 {c}^{3} + 6 {c}^{2} - 18 c + 10\right)$ with $c = - 5$
If the result is $0$ then $\left(c + 5\right)$ is a factor of the given polynomial

#### Explanation:

The Factor Theorem say:

$\left(x - a\right)$ is a factor of $f \left(x\right)$ if and only if $f \left(a\right) = 0$

The Remainder Theorem says

$f \left(a\right)$ is the remainder of $f \frac{x}{x - a}$

For the given expressions this means

$\left(c + 5\right)$ is a factor of ${c}^{4} + 7 {c}^{3} + 6 {c}^{2} - 18 c + 10$

if and only if $\frac{\textcolor{red}{1} {c}^{4} + \textcolor{red}{7} {c}^{3} + \textcolor{red}{6} {c}^{2} \textcolor{red}{- 18} c + \textcolor{red}{10}}{c - \textcolor{b l u e}{\left(- 5\right)}} = \textcolor{g r e e n}{0}$

Applying synthetic division:

{: (," | ",color(red)(1),color(white)("X")color(red)(7),color(white)("XX")color(red)(6),color(red)(-18),color(white)("X")color(red)(10)), (color(blue)(-5)," | ",,-5,-10,color(white)("X") 20,-10), ("---","---","---","-----","------","------","-------"), (,,1,color(white)("X")2,color(white)("X")-4,color(white)("XX")2,color(white)("XX")color(green)(0)) :}

Giving a remainder of $\textcolor{g r e e n}{0}$
$\Rightarrow \left(c - \left(- 5\right)\right) = \left(c + 5\right)$ is a factor of ${c}^{4} + 7 {c}^{3} + 6 {c}^{2} - 18 c + 10$