# Rational Expressions

**Operations with Rational Expressions **

The operations of addition, subtraction, multiplication, and division with
rational expressions follow the same

rules that are used with common fractions. The key, as with simplifying rational
expressions, is to factor the

polynomials .

**I. Multiplication of Rational Expressions**

Recall with fractions
Also recall when mulitplying fractions that you may

cross-cancel; that is, reduce common factors of any numerator with any
denominator.

**Example 1:**

The same is true of multiplying rational expressions. As before, the key is to factor the polynomials first.

**Example 2: **Simplify

Solution : First factor the polynomials. Don’t forget to cancel common factors!

Before looking at the next example, recall our strategies
for factoring:

1. Factor out any common factors

2. If more than 3 terms, try factoring by grouping

3. Recognize special products

4. Factor trinomials using product /sum strategies:

a) factors into (x+m)(x+n) where

mn=c and m+n=b

b) split bx into mx+nx, where

mn=ac and m+n=b, then factor by grouping.

**Example 3:** Simplify

Solution:

**Example 4: **Simplify

Solution:

## II. Division of Rational Expressions

Recall with fractions
In other words, dividing by a fraction is the

same as multiplying by its reciprocal. The same is true for rational
expressions. As before, the key to making

the work easier is to factor the polynomials first.

**Example 5:**

Solution: Begin by writing the second term as

Notice that the x+2 terms cannot be cancelled since both are in the denominator.

Multiplication and Division can also appear together. Only
take the reciprocal of the fractions you are

dividing.

**Example 6:**

Solution: We only take the reciprocal of the second fraction:

**Question:** Explain why the values -7, -6, 3, and 4
are excluded in the previous example.

Answer: The easiest way to understand why these values have been excluded is to
write

y≠-6 follows since 4y+24 cannot equal 0.

We also get y≠3 because the term

The next observation is that the denominator of

that is,cannot equal
zero. Factoring the numerator and denominator of this rational expression

we have

The numerator is zero if y =-7 or if y=4. This accounts for
excluding the numbers -7 and 4. The

denominator of

factors into which means we have to exclude -6, but we have already done that.

**Question:** What values of x are not allowed in the
rational expression

Answer: 3, 4, 6 (The value 3 is not allowed because
is zero at x=3 and we cannot divide by 0.

x=4 is not allowed because we have the term x-4 in the denominator. Similarly
x=6 is not allowed

because the term x-6 is the denominator of

## III. Addition and Subtraction of Rational Expressions

Recall that addition and subtraction is done by first
finding a common denominator. The least common

denominator (LCD) of several fractions is the product of all prime factors of
the denominators. A factor only

occurs more than once in the LCD if it occurs more than once in any one
fraction. Once you have the LCD,

convert each fraction to an equivalent fraction with the LCD, then add or
subtract the numerators.

**Example 7:**

Solution: The LCD here is
We convert each fraction to an equivalent
one with this

denominator: That is,

**Example 8:**

Solution: We must factor the denominators first to find the LCD:

The LCD is We now convert each fraction to an equivalent fraction using this denominator:

Next subtract the two expressions

Note that the minus sign in the numerator was distributed across the expression

**Question:** What is the common denominator in

Answer:

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