# Algebra: Adding and Subtracting Polynomials

## Adding and Subtracting Polynomials

In previous sections, you've simplified expressions like 3*x* + 7*x* to get 10*x*, or perhaps subtracted terms like 5*y* - 9*y* to get -4*y*. That arithmetic makes perfect sense if you translate the mathematics into words. For example, the expression 3*x* + 7*x* literally means "three of a certain number added to seven more of that same number," which is definitely equal to "10 of that number," or 10*x*.

##### Talk the Talk

**Like terms** have variables which match exactly, like 4*x*^{2}*y*^{3} and -7*x*^{2}*y*^{3}. You can only add or subtract two terms if they are like terms.

What I didn't tell you back then was that you were only allowed to combine the coefficients of those terms because they contained *the exact same variables*. Actually, any two terms whose variables match *exactly* are called *like terms*, and you cannot add terms together or subtract them from one another if they are not like terms.

##### Kelley's Cautions

Many students try to simplify the expression 4*x* + 5*y* and get 9*xy*, but that's wrong! Remember, you can't add or subtract 4*x* and 5*y* because the terms have different variables. It would be like adding four cats to five dogs and getting nine dats (or cogs). Unlike terms are like apples and oranges—you can't combine them.

If two terms have the same variables and get all nervous when they look at each other, you can upgrade them from like terms to love terms, but since it's hard to read the emotions of variables (they're always changing on you), most mathematicians don't even try to make that differentiation.

Now you know why the expression 13*x*^{2}*y*^{3} - 5*x*2*y*^{3} can be simplified as 8*x*^{2}*y*^{3}. Since the variables in both terms match exactly (they both contain *x*^{2}*y*^{3}), all you have to do is combine the coefficients and attach a copy of the matching variable string.

**Example 2**: Simplify the following expression.

- 4
*x*^{3}+ 5*x*^{2}- 3*x*+ 1 - (2*x*^{3}- 8*x*^{2}+ 9*x*- 6)

**Solution**: Start by applying the distributive property (multiply everything in the parentheses by -1).

- 4
*x*^{3}+ 5*x*^{2}- 3*x*+ 1 - 2*x*^{3}+ 8*x*^{2}- 9*x*+ 6

##### You've Got Problems

Problem 2: Simplify the expression 3*x*^{2} - 9 + 2(*x*^{2} - 6*x* + 5).

If you rewrite the expression so that like terms are grouped together, it makes simplifying easier.

- 4
*x*^{3}- 2*x*^{3}+ 5*x*^{2}+ 8*x*^{2}- 3*x*- 9*x*+ 1 + 6

Combine the coefficients of each pair of like terms.

- 2
*x*^{3}+ 13*x*^{2}- 12*x*+ 7

Excerpted from The Complete Idiot's Guide to Algebra © 2004 by W. Michael Kelley. All rights reserved including the right of reproduction in whole or in part in any form. Used by arrangement with **Alpha Books**, a member of Penguin Group (USA) Inc.

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