# Expressions

## Rationalize Denominator in Rational Expressions

Fractions show parts of wholes. The numerator tells how many pieces a whole is cut into and the numerator tells how many of those pieces you have.

## Dividing Rational Expressions

Because rational expressions are just fractions with variables, the process of dividing rational expressions is exactly the same as the process of dividing fractions. You just find the reciprocal of the second rational expression (flip it, so the the denominator becomes the numerator and the numerator becomes the denominator), and then multiply the rational expressions (numerator times numerator and denominator times denominator).  It's really that easy!

## Multiply Rational Expressions

Because rational expressions are just fractions with variables, the process of multiplying rational expressions is exactly the same as the process of multiplying fractions. Multiply the rational expressions just like you multiply fractions (numerator times numerator and denominator times denominator).

## Adding & Subtracting Rational Expressions

Rational expressions are fractions that have variables (including binomals or polynomials) in them.  The variables can be in the numerator or the denominator (or both!).

The rules for adding and subtracting rational expressions are exactly like those for adding and subtracting fractions.  If the denominators are the same, you can add or subtract the numerators.  If the denominators are different, you have to find a common denominator before you can add or subtract the numerators.

## Division with Polynomials

Division with polynomials looks complicated.  But you can do long division with polynomials in much the same way that you do division with numbers!  And, like with long division, you just have to pay attention to where you put your answer terms so that you can keep track of where you are in the problem.

To do polynomial division, you use the long division "house."  Let's review that process very quickly:

The problem: $56 \div 4=$ is written: $4\overline{)56}$

## Multiplying Binomials (FOIL)

When you multiply polynomials, you must make sure that you multiply each term by each other term.  When multiplying binomials, we ensure that each term is multiplied by each other term using the FOIL process.

Foil is an acronym that stands for first, outer, inner, last.

The basic process of using foil, is to take the two binomials and first multiply the "first" terms of each binomial together.

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## Multiplying Polynomials

Sometimes algebra problems will ask you to combine polynomials with multiplication.  These problems can look intimidating, but they are very similar to multiplying binomials (remember FOIL?).  The secret to these problems is being methodical and making sure that every term in each polynomial is multiplied by every term in each of the other polynomials.

Remember that:

In algebra problems, you will often have to combine polynomials with addition or subtraction. These problems can look confusing, but they are fairly simple. You just have to do them carefully, paying particular attention to negative signs.

There is an associative property of addition, so you can remove the parentheses and just combine the like terms from each polynomial.  Pay chose attention to:

## Multiply Variable Expressions

We all know how to multiply, but sometimes multiplying expressions that contain variables can get confusing.  The main point you want to remember is that multiplication works the same, no matter what you are multiplying:

## Undefined

Fractions can be undefined. An undefined fraction is a fraction that doesn't make sense.  Simply: an undefined fraction is a fraction with a zero denominator.

If a have a pizza and cut it into 6 pieces, I can give you 1 piece ($\dfrac{1}{6}$) or 4 pieces ($\dfrac{4}{6}$).