8.2 Multiply and Divide Rational Expressions

Multiplying Rational Expressions

Multiplying and dividing rational expressions works a lot like multiplying and dividing regular fractions. The key difference is that you're dealing with polynomials instead of plain numbers, so factoring becomes your best friend.

A rational expression is a fraction where both the numerator and denominator are polynomials. For example, $\frac{x^2 - 4}{x + 3}$ is a rational expression. Before doing any operation, always check for domain restrictions: values of the variable that would make any denominator equal zero. Those values are excluded from the domain, even if they cancel out later.

Multiplication of Rational Expressions

The process mirrors regular fraction multiplication, but you'll want to factor before you multiply across. This saves you from wrestling with large polynomials at the end.

Steps to multiply rational expressions:

1. Factor every numerator and denominator completely.
2. Write the multiplication as a single fraction (numerators on top, denominators on bottom).
3. Cancel any factors that appear in both the numerator and denominator.
4. Multiply the remaining factors to get your simplified answer.

Example: Simplify $\frac{2x}{3} \cdot \frac{3x}{4}$

1. Nothing to factor here.
2. Write as one fraction: $\frac{2x \cdot 3x}{3 \cdot 4} = \frac{6x^2}{12}$
3. Cancel the common factor of 6: $\frac{x^2}{2}$

Example with polynomials: Simplify $\frac{x^2 - 4}{x + 3} \cdot \frac{x + 3}{x + 2}$

1. Factor: $\frac{(x+2)(x-2)}{x+3} \cdot \frac{x+3}{x+2}$
2. Write as one fraction: $\frac{(x+2)(x-2)(x+3)}{(x+3)(x+2)}$
3. Cancel $(x+2)$ and $(x+3)$: $\frac{\cancel{(x+2)}(x-2)\cancel{(x+3)}}{\cancel{(x+3)}\cancel{(x+2)}}$
4. Result: $x - 2$

Remember the product rule of exponents when multiplying variables: $x^2 \cdot x^3 = x^5$ (add the exponents).

Dividing Rational Expressions

Division of Rational Expressions

Division works exactly like multiplication with one extra step at the beginning: flip the second fraction (take its reciprocal), then multiply.

Steps to divide rational expressions:

1. Rewrite the division as multiplication by the reciprocal of the second expression.
2. Factor every numerator and denominator completely.
3. Cancel common factors.
4. Multiply the remaining factors.

Example: Simplify $\frac{2x}{3} \div \frac{4}{5x}$

1. Flip the second fraction: $\frac{2x}{3} \cdot \frac{5x}{4}$
2. Multiply across: $\frac{10x^2}{12}$
3. Cancel the common factor of 2: $\frac{5x^2}{6}$

Note: $x^2$ and $6$ share no common factors, so $\frac{5x^2}{6}$ is the final answer. (You can't cancel the $x^2$ with the 6.)

Complex Fractions with Rational Expressions

A complex fraction has fractions in its numerator, its denominator, or both. To simplify one, multiply the top and bottom by the LCD (least common denominator) of every smaller fraction inside it.

Example: Simplify $\frac{\frac{2}{x}}{\frac{3}{x^2}+\frac{1}{x}}$

1. Find the LCD of all the inner denominators. The denominators are $x$ and $x^2$, so the LCD is $x^2$.

2. Multiply every term in the numerator and denominator by $x^2$:

   • Numerator: $\frac{2}{x} \cdot x^2 = 2x$
   • Denominator: $\frac{3}{x^2} \cdot x^2 + \frac{1}{x} \cdot x^2 = 3 + x$

3. Write the simplified result: $\frac{2x}{3+x}$

4. Check for common factors. Here $2x$ and $3+x$ share none, so this is the final answer.

Working with Rational Expressions

A few things to keep in mind across all these problems:

• Factor first, multiply second. Factoring before you multiply makes canceling much easier and keeps the numbers manageable.
• Never cancel terms that are added or subtracted. You can only cancel factors (things being multiplied). For instance, in $\frac{x + 3}{3}$, you cannot cancel the 3s because the numerator's 3 is being added, not multiplied.
• State domain restrictions. If $x + 3$ appeared in a denominator before you canceled it, then $x \neq -3$ is still a restriction on the original expression.
• Always simplify to lowest terms. After canceling, check that the numerator and denominator share no remaining common factors.