Tips for Multiplying And Dividing Rational Expressions

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Rational expressions seem more complicated than basic integers, but the rules for multiplying and dividing them are easy to understand. Whether you are tackling a complicated algebraic expression or dealing with a simple fraction, the rules for multiplication and division are basically the same. After you learn what rational expressions are and how they relate to ordinary fractions, you will be able to multiply and divide them with confidence.

TL;DR (Too Long; Didn't Read)

Multiplying and dividing rational expressions works just like multiplying and dividing fractions. To multiply two rational expressions, multiply the numerators together, and then multiply the denominators together.

To divide one rational expression by another, follow the same rules as dividing one fraction by another. First, turn the fraction in the divisor (which you divide by) upside down, and then multiply it by the fraction in the dividend (which you are dividing).

What Is a Rational Expression?

The term “rational expression” describes a fraction where the numerator and denominator are polynomials. A polynomial is an expression like

2x^2 + 3x + 1

composed of constants, variables and exponents (that are not negative). The following expression:

\frac{x + 5}{x^2 - 4}

Provides an example of a rational expression. This basically has the form of a fraction, just with a more complicated numerator and denominator. Note that rational expressions are only valid when the denominator does not equal zero, so the example above is only valid when ​x​ ≠ 2.

Multiplying Rational Expressions

Multiplying rational expressions follows basically the same rules as multiplying any fraction. When you multiply a fraction, you multiply one numerator by the other and one denominator by the other, and when you multiply rational expressions, you multiply one whole numerator by the other numerator and the whole denominator by the other denominator.

For a fraction you write:

\begin{aligned} \frac{2}{5} × \frac{4}{7} &= \frac{2 × 4}{5 × 7} \\ \,\\ &= \frac{8}{35} \end{aligned}

For two rational expressions, you use the same basic process:

\begin{aligned} \frac{x + 5}{x - 4} × \frac{x}{x + 1} &= \frac{(x + 5) × x}{(x - 4) × (x + 1)} \\ \,\\ &= \frac{x^2 + 5x}{x^2 -4x + x - 4} \\ \,\\ &= \frac{x^2 + 5x}{x^2 - 3x - 4} \end{aligned}

When you multiply a whole number (or algebraic expression) by a fraction, you simply multiply the numerator of the fraction by the whole number. This is because any whole number ​n​ can be written as ​n​ / 1, and then following the standard rules for multiplying fractions, the factor of 1 does not change the denominator. The following example illustrates this:

\begin{aligned} \frac{x + 5}{x^2 - 4} × x &= \frac{x + 5}{x^2 - 4} × \frac{x}{1} \\ \,\\ &= \frac{(x + 5) × x}{(x^2 - 4) × 1}\\ \,\\ =& \frac{x^2 + 5x}{x^2 - 4} \end{aligned}

Dividing Rational Expressions

Like multiplying rational expressions, dividing rational expressions follows the same basic rules as dividing fractions. When you divide two fractions, you turn the second fraction upside down as the first step, and then multiply. So:

\begin{aligned} \frac{4}{5} ÷ \frac{3}{2} &= \frac{4}{5} × \frac{2}{3} \\ \,\\ &= \frac{4 × 2}{5 × 3} \\ \,\\ &= \frac{8}{15} \end{aligned}

Dividing two rational expressions works in the same way, so:

\begin{aligned} \frac{x + 3}{2x^2} ÷ \frac{4}{3x} &= \frac{x + 3}{2x^2} × \frac{3x}{4} \\ \,\\ &= \frac{(x + 3) × 3x}{2x^2 × 4} \\ \,\\ &= \frac{3x^2 + 9x}{8x^2} \end{aligned}

This expression can be simplified, because there is a factor of ​x​ (including ​x2) in both terms in the numerator and a factor of ​x2 in the denominator. One set of ​x​s can cancel to give:

\begin{aligned} \frac{3x^2 + 9x}{8x^2} &= \frac{x(3x + 9)} {8x^2} \\ &= \frac{3x + 9}{8x} \end{aligned}

You can only simplify expressions when you can remove a factor from the whole expression on the top and bottom as above. The following expression:

\frac{x - 1}{x}

Cannot be simplified in the same way because the ​x​ in the denominator divides the whole term in the numerator. You could write:

\begin{aligned} \frac{x-1}{x} &= \frac{x}{x} - \frac{1}{x} \\ &= 1 - \frac{1}{x} \end{aligned}

If you wanted to, though.

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