Simplifying Rational Expressions

When dealing with rational equations, it comes in handy to simplify the expressions in order to identify some of the characteristics of these rational functions, and to manipulate these expressions.

In order to simplify rational expressions, factor the numerator and denominator first. Then eliminate all of the multiplicative identity pairs to simplify.

Example 1:

\frac{60}{126}

\frac{2 • 2 • 3 • 5}{2 • 3 • 3 • 7}

Expand the numerator and denominator with prime factorization.

\frac{2 • 2 • 3 • 5}{2 • 3 • 3 • 7}

Simplify by eliminating all multiplicative identity pairs.

\frac{2 • 5}{3 • 7} = \frac{10}{21}

Rewrite in simplest terms.

Example 2:

\frac{7 x^{5} y^{3}}{21 x^{2} y^{4}}

\frac{7 • x • x • x • x • x • y • y • y}{3 • 7 • x • x • y • y • y • y}

Expand the numerator and denominator.

\frac{7 • x • x • x • x • x • y • y • y}{3 • 7 • x • x • y • y • y • y}

Simplify by eliminateing identity pairs.

\frac{x^{3}}{3 y}

Rewrite in simplest terms.

Example 3:

\frac{4 x^{2}}{24 x^{2} - 8 x}

\frac{4 x • x}{4 x (6 x - 2)}

Expand the numerator and factor the denominator.

\frac{4 x • x}{4 x (6 x - 2)}

Simplify.

\frac{x}{6 x - 2}

Rewrite.

Sometimes the rational expression is already in simplest form.

Example:

\frac{5}{x (x + 3)}

There are no common factors between 5 and x(x + 3), so it cannot be simplified.

Simplify by factoring binomials

Example 4:

\frac{30 x + 15}{5 x + 20}

\frac{15 (x + 3)}{5 (x + 4)}

Factor the numerator and denominator by their respective GCFs.

\frac{3 • 5 (x + 3)}{5 (x + 4)}

Simplify.

\frac{3 (x + 3)}{x + 4}

Rewrite.

Example 5:

\frac{x^{2} - 11 x + 24}{x^{2} - 3 x - 40}

\frac{(x - 8) (x - 3)}{(x - 8) (x + 5)}

Factor the numerator and denominator.

\frac{(x - 8) (x - 3)}{(x - 8) (x + 5)}

Simplify.

\frac{x - 3}{x + 5}

Rewrite.

Example 6:

\frac{x^{3} - 5 x^{2} - 3 x + 15}{x^{2} - 8 x + 15}

\frac{x^{2} (x - 5) - 3 (x - 5)}{(x - 5) (x - 3)}

Factor four terms by factoring two at a time

\frac{(x^{2} - 3) (x - 5)}{(x - 5) (x - 3)}

The other factor follows.

\frac{(x^{2} - 3) (x - 5)}{(x - 5) (x - 3)}

Simplify.

\frac{x^{2} - 3}{x - 3}

Rewrite.

Now that you know how to simplify rational expressions, you will be able to perform addition and subtraction with rational expressions.

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