Lots of times you’re going to have to do this crazy thing called “combining like terms.” I know it might seem like some arbitrary procedure invented by mathematicians to make you want to pull you hair out, but it’s actually super easy.

Remember our old friend the distributive law? For all numbers a, b, and c, the following is true: a(b+c)=ab+ac? Of course you do. Combining like terms is just the application of the distributive law. say we’ve got this problem, and the instructions are to “simplify”.

2x^{2}+3x^{2}

We use the distributive law, with x^{2} playing the role of “a”, 2 playing the role of “b” and 3 playing the role of “c”.

2x^{2}+3x^{2}=(2+3)x^{2}

And you know what 2+3 is, right? Right.

2x^{2}+3x^{2}=(2+3)x^{2}=5x^{2}

How many x squareds do we have? 5! Easy peasy.

If you have a problem that’s a little more complicated, the same method applies.

2x^{2}+3x+4x^{2}+5x

First we rearrange terms using the commutative law of addition, so all the x squareds are with the x squareds and all the x’s are with all the x’s:

2x^{2}+3x+4x^{2}+5x

=2x^{2}+4x^{2}+3x+5x

Now do the same distributive law magic we did before:

2x^{2}+3x+4x^{2}+5x

=2x^{2}+4x^{2}+3x+5x

=(2+4)x^{2}+(3+5)x

Now add!

=6x^{2}+8x

We can take a shortcut too… if you don’t want to write out all those steps, that’s okay. If we have a look at 2x^{2}+3x+4x^{2}+5x again, paying attention to coefficients (that’s the numbers in front of x^{2} or x), you can see that the coefficients of x^{2} are 2 and 4, so add ‘em. That’s your new coefficient for x^{2}. The coefficients of x are 3 and 5, we add those and get 8, that’s our new coefficient of x. So our new expression, after combining like terms, is =6x^{2}+8x, just like before.

PROTIP: x^{2} and x ARE NOT like terms. The exponent on the variable has to match to have like terms. We can’t combine 2x^{2}+3x into 5x^{2} or 5x^{3} or whatever. Don’t do it. You’ll make me cry.

Check out this video for more combining like terms goodness.

http://www.brightstorm.com/math/algebra/pre-algebra/simplifying-expressions-and-combining-like-terms-problem-1/