# 2.7: Algebra Tips and Tricks Part III (Factoring)

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

## 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 $$x^2 - 64 = (x + 8)(x - 8)$$.

This page titled 2.7: Algebra Tips and Tricks Part III (Factoring) is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Tyler Seacrest via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.