Simplifying Expressions

To simplify an expression means to write an equivalent expression which contains no similar terms.  This means that we will rewrite the expression with the fewest terms possible.

When faced with an expression like $4x+5\left(3x-12\right)$ , what do we do first? Let's see: PEMDAS says work in parentheses first, but $3x$ and $12$ are unlike terms . Hmm, let's try the distributive property :

$\begin{array}{l}4x+5\left(3x-12\right)\\ =4x+5\left(3x\right)+5\left(12\right)\\ =4x+15x-60\\ =19x-60\end{array}$ .

This problem was no problem!

What about $\left(4x+5\right)\left(3x-12\right)$ ? Is this the same as $4x+5\left(3x-12\right)$ ?

No, the parentheses change it. Here we can use the distributive property twice:

$\begin{array}{l}\left(4x+5\right)\left(3x-12\right)\\ =\left(4x+5\right)\left(3x\right)-\left(4x+5\right)\left(12\right)\end{array}$

$=12{x}^{2}+15x-48x-60$ (remember to change the sign on that last term)

$=12{x}^{2}-33x-60$ .

That worked, but it was long. Is this the only way? No. The best way? No. Use the " FOIL system":

F irst, O utside, I nside, L ast.

$\left(4x+5\right)\left(3x-12\right)$

$=\stackrel{\text{First}}{\stackrel{︷}{\left(4x\right)\left(3x\right)}}-\stackrel{\text{Outside}}{\stackrel{︷}{\left(4x\right)\left(12\right)}}+\stackrel{\text{Inside}}{\stackrel{︷}{\left(5\right)\left(3x\right)}}-\stackrel{\text{Last}}{\stackrel{︷}{\left(5\right)\left(12\right)}}$

$=12{x}^{2}-48x+15x-60=12{x}^{2}-33x-60$ .

Better!

Here's another example:

$\left(n+3\right)\left(n-3\right)={n}^{2}-3n+3n-9={n}^{2}-9$ .

Notice the "middle terms" cancel, and we're left with what's called the difference of two squares.

In general, $\left(a+b\right)\left(a-b\right)={a}^{2}-{b}^{2}$ . Also see the factoring section.