# Algebraic Expressions

### Topics

You probably don't keep bananas in your sock drawer or toothpaste on your bookshelf. If you do, we can recommend a good therapist. Aside from it being an awful hassle to run from the bookshelf to the bathroom sink every time you need to spit, the reason we don't do such things is because it makes sense to keep like things together. Algebraic terms are no exception.

A term in an expression has two parts: the constant part, also called the **coefficient**, where this word describes a term equally as efficiently as the word "constant," so it is "coefficient," and the variable part. Just as they sound, the constant part is the product of all the constants in the term, and the variable part is the product of all the variables in the term. If someone approaches you about Variable Life and Annuities, they probably want your help on an algebra problem.

### Sample Problems

1. In the term 9*x*, the constant part, or coefficient, is 9 and the variable part is *x*.

2. In the term 2*y*(3*x*^{2}), the coefficient is 6 and the variable part is *x*^{2}*y*.

3. In the term *mc*^{2} where *c* is a constant, *c*^{2} is the coefficient and *m* is the variable part.

In an expression, **like terms** are terms with the same variable part. Going back to one of our earlier examples, bananas go with bananas, and socks go with socks. Although, if bananas *could* go with socks, we sure could make some hilarious puppets.

### Sample Problems

1. *x* and 8*x* are like terms, since they both have *x* as their variable part.

2. 5*x*^{7}*y* and 89*x*^{7}*y* are like terms, since they both have *x*^{7}*y* as their variable part.

3. 4 and 5 are like terms, since both are constants with no variable part. We know what we are getting *them* for Chanukah.

### Non-Sample Problems

1. *x* and *x*^{2} are not like terms, since they have different variable parts.

2. *y* and *x* are not like terms, since they have different variable parts.

3. *x*^{2}*y* and *xy*^{2} are not like terms, since they have different variable parts.

If we have a pile with 5 lemons and a pile with 6 lemons, we can combine them to get one pile with 11 lemons. Of course, what we actually should do is make lemonade, but that is more of a life philosophy and less of a mathematical solution. In this real-life situation—you may often be confronted with such a fruity scenario—the lemons are our variables. Therefore, we can **combine like terms**. Or, in this case, combine like lemons.

### Sample Problems

1. Combining 3 homework assignments with 2 homework assignments gives us 5 homework assignments. Plus 1 disgruntled student.

2. Combining 9 fingers and 1 finger gives us 10 fingers. Thank goodness. We thought we misplaced one for a second there.

3. Combining 3 copies of the letter *x* with 5 copies of the letter *x* gives us 8 of the letter *x*. With which we can spell "xxxxxxxx."

In order to combine terms, they need to be like terms. We can't combine *x* and 2*x*^{2}. We could use the distributive property to rewrite *x* + 2*x*^{2} as (1 + 2*x*)*x* (give it a try; the math checks out), but (1 + 2*x*)*x* isn't that much prettier than *x* + 2*x*^{2}. Not to sound superficial, but when it comes to mathematical expressions, it is all about the looks.

### Sample Problems

1. Combine like terms in the expression 7*x* + 3*xy* + 3 + 4*xy* + 2*x*^{2} + 6.

First, rearrange the expression to put like terms next to each other. Pull your bananas out of your sock drawer and move them over to the banana drawer of your dresser. Wait a minute...

7*x* + 3*xy* + 4*xy* + 2*x*^{2} + 3 + 6. Now it is easier to see what goes with what. Which is good, because you don't want your terms to clash.

Combine the terms with variable part *xy* to get 7*x* + 7*xy* + 2*x*^{2}, and then combine the constant terms to get 7*x* + 7*xy* + 2*x*^{2} + 9.

This line is as clean and clear as it gets. Without over-the-counter acne medication, anyway.

**Be careful** when fractions are involved. Some of them have sharp edges. Using the distributive property to combine like terms with fractional coefficients is a little more complicated; therefore, pay extra close attention to what you are doing. Put on your thinking cap, if you have one. If you don't, a particularly snug baseball cap will do.

2. Combine like terms:

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First, do some rearranging such that like terms are next to each other:

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Next, use the distributive property to factor out the variables.

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Finally, add the fractions as we learned to do in the section on fractions. As we are sure you recall, in order to perform the arithmetic you will need to convert these guys so that they have like denominators. If you don't remember how to do that, return to Start and draw 2 cards.

Thinking about the distributive property allows us to separate out the coefficients from the variables; therefore, we don't need to think about both at once. Remember how well that worked when you were trying to think at the same time about riding your bicycle and about that kiss you got from Heather Pelnicki? Didn't turn out well for you, did it? Didn't turn out well for Heather Pelnicki, either. To be fair, she shouldn't have been standing right in the middle of the bike lane.