Summary For Rational Expressions and Equations

Study Guide: Rational Expressions and Equations

Rational expressions and equations are algebraic expressions involving fractions, where the
numerator and denominator are polynomials. Understanding how to simplify, manipulate, and
solve these types of expressions and equations is vital for mastering algebra.

1. Rational Expressions

A rational expression is an expression in the form of a fraction, where both the numerator and
the denominator are polynomials. For example:

3x2+2x−5x2−4\frac{3x^2 + 2x - 5}{x^2 - 4}

Here, both 3x2+2x−53x^2 + 2x - 5 and x2−4x^2 - 4 are polynomials.

To simplify rational expressions, factor both the numerator and denominator and cancel out any
common factors. For example, consider:

x2−9x2−6x+9\frac{x^2 - 9}{x^2 - 6x + 9}

The numerator factors as (x−3)(x+3)(x - 3)(x + 3), and the denominator factors as (x−3)2(x -
3)^2. Simplifying, we get:

x+3x−3\frac{x + 3}{x - 3}

This is the simplified form.

2. Domain of Rational Expressions

The domain of a rational expression consists of all possible values of xx for which the
expression is defined. Since division by zero is undefined, we must exclude values of xx that
make the denominator zero.

For example, in the expression 3xx2−4\frac{3x}{x^2 - 4}, the denominator factors as
(x−2)(x+2)(x - 2)(x + 2), so the expression is undefined for x=2x = 2 and x=−2x = -2. Therefore,
the domain is all real numbers except x=2x = 2 and x=−2x = -2.

3. Simplifying Rational Expressions

To simplify a rational expression:

1.​ Factor both the numerator and denominator completely.
2.​ Cancel out any common factors from the numerator and denominator.
3.​ Simplify the result by rewriting the expression without the canceled factors.

Example: Simplify the expression 2x2−84x2−16\frac{2x^2 - 8}{4x^2 - 16}.