# How do you factor y= x^3 + 4x^2 - x - 4 ?

Feb 28, 2016

$\left(x + 4\right) \left(x + 1\right) \left(x - 1\right)$

#### Explanation:

We are given $y = {x}^{3} + 4 {x}^{2} - x - 4$. Because the highest exponent is $3$, we know there are $3$ solutions.

With anything larger than a quadratic ($a {x}^{2} + b x + c$), I like to use synthetic division. To use synthetic division, the first thing we must do is find the divisor. We can do that by guessing-and-checking, or we can graph the equation and use the roots (x-intercepts) as our divisor. Let's do that right now:
graph{y=x^3+4x^2-x-4}

Lucky for us, we can see all three solutions. I'm still going to use synthetic division just to show you how it works:

$\textcolor{w h i t e}{- 1.} |$$\textcolor{w h i t e}{.} 1$$\textcolor{w h i t e}{\ldots} 4$$\textcolor{w h i t e}{.} - 1$$\textcolor{w h i t e}{.} - 4$
$- 1 \textcolor{w h i t e}{.} |$_$- 1$__$- 3$__$4$_
$\textcolor{w h i t e}{\ldots \ldots \ldots} 1 \textcolor{w h i t e}{\ldots} 3 \textcolor{w h i t e}{. .} - 4 \textcolor{w h i t e}{\ldots .} 0$

This can be rewritten as ${x}^{2} + 3 x - 4$, which can be factored to $\left(x - 1\right) \left(x + 4\right)$.

So, we have $\left(x + 4\right) \left(x - 1\right) \left(x + 1\right)$! And that's what the graph shows too. Nice work!