# Rational Expressions Review

Recall: A rational number is any number that can be
expressed in the form a/b where a and b are

integers with b not zero .

A rational expression is an expression p/Q where P and Q are polynomials.

**Where is a rational expression undefined?
**Where is undefined? Recall: Any fraction
is undefined when the denominator is 0. So we must

find the value for x that makes the denominator 0 to see where the rational expression is undefined.

x – 6 = 0 x = 6 So the expression is undefined for x = 6

Where is undefined?

Solve x ^{2} – 3x – 10 = 0

(x – 5)(x + 2) = 0

x – 5 = 0 or x + 2 = 0 The rational expressions is undefined

when x = 5 or x = – 2

What about If we
try to solve x^{2} + 2 = 0, we see that there is no solution. Thus, this

expression is defined for all real numbers since there is no real number value
for x that makes the

denominator 0.

** Simplifying Rational Expressions
Property :**

Multiplying or dividing both the numerator and denominator by the same nonzero quantity will not

change the value . (Just like with regular fractions)

** Rule : **To reduce a rational expression to lowest
terms, factor the numerator and denominator

completely and divide out any factors they have in common.

Examples:

Factor the numerator : x(x^{2} – x –
2) = x(x – 2)(x + 1)

Factor the denominator: x(x^{2} + 4x + 3) = x(x + 3)(x + 1)

So we have:

**When factors are opposites**

Consider: The numerator
and denominator * are not* the same. Note − (2 – x) = − 2 + x = x – 2

So the numerator and denominator are opposites. We can write the rational
expression as:

Simplify:

**Multiplication and Division**

Recall: When we
multiply fractions, we factor and divide out common factors .

We do the same with rational expressions except we may be factoring polynomials
as well as numbers.

Example:

Example:

__ Division__Recall: Dividing by a number is the same as multiplying by its
reciprocal

**Exercises for Chapter 6 Review**

Reduce the following rational expressions to lowest terms. State any restrictions on the variable.

Reduce the following rational expressions to lowest terms.

Multiply or divide as indicated.

**Solutions for Chapter 6 Review Exercises**

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