Factoring

When factoring an expression like this:

$x^2-8x+15$

The goal is to write this like $(x+a)(x+b)$ for some numbers $a$ and $b$, where $a$ and $b$ could be positive, negative, or zero. Since $(x+a)(x+b)=x^2+(a+b)x+(ab)$ we see we need $a+b=-8$ and $ab=15$. That way, when you foil it back out, you have $x^2-8x+15$. We see if $a=-3$ and $b=-5$, this works for both $a+b=-8$ and $ab=15$. Thus,

$x^2-8x+15=(x-3)(x-5)$

Let’s do a couple more examples.

• Factor $x^2+3x+2$.

  In this case we want $a+b=3$ and $ab=2$. $a=1$ and $b=2$ works, so $x^2+3x+2=(x+1)(x+2)$.

• Factor $x^2+5x-84$.

  This is a bit harder because the numbers are bigger, but we can still do it. We want $a+b=5$, and $ab=-84$. We can see that $84$ is $12$ times $7$. So if we have $a=12$ and $b=-7$, then $a+b=5$ and $ab=-84$. Hence $x^2+5x-84=(x+12)(x-7)$.

• Factor $x^2-64$.

  In this case, we want $a+b=0$ and $ab=-64$. But notice that this means $a=-b$, and hence $-a^2=-64$, which means $a^2=64$. That means $a=8$, so $b=-8$ (or vice versa). Hence $x^2-64=(x+8)(x-8)$.