8.1 Simplify Rational Expressions

Simplifying Rational Expressions

Introduction to Rational Expressions

A rational expression is a fraction where the numerator and denominator are both polynomials. Think of it exactly like a numeric fraction such as $\frac{3}{4}$, except now you have variables involved, like $\frac{x+2}{x^2-1}$.

Simplifying rational expressions means reducing them to their simplest form, just like you'd reduce $\frac{6}{8}$ to $\frac{3}{4}$. The process relies on factoring and canceling common factors. Before simplifying, though, you always need to identify variable restrictions, the values of the variable that would make the denominator zero.

Undefined Values in Rational Expressions

A rational expression is undefined whenever its denominator equals zero (division by zero is not allowed). To find the restricted values, set the denominator equal to zero and solve.

For example, with $\frac{2x+1}{x-5}$:

1. Set the denominator equal to zero: $x - 5 = 0$

2. Solve: $x = 5$

So $x = 5$ is a restricted value. You should always state these restrictions alongside your simplified answer.

If the denominator is something like $x^2 - 4$, factor it first to $(x+2)(x-2)$, then set each factor to zero. That gives you $x = -2$ and $x = 2$ as restricted values.

Evaluating and Simplifying Rational Expressions

Evaluating a rational expression means plugging in a specific value for the variable and computing the result. If that value makes the denominator zero, the expression is undefined at that point.

Simplifying follows the same logic as reducing numeric fractions:

1. Factor the numerator and denominator completely.
2. Cancel any factors that appear in both the numerator and denominator.
3. Write the simplified expression using the remaining factors.

Example: Simplify $\frac{x^2-9}{x+3}$

1. Factor the numerator: $x^2 - 9 = (x+3)(x-3)$, so the expression becomes $\frac{(x+3)(x-3)}{x+3}$

2. Cancel the common factor $(x+3)$

3. Simplified result: $x - 3$, with the restriction $x \neq -3$

Notice that even though $(x+3)$ canceled, you still note that $x = -3$ is restricted because it made the original denominator zero.

Simplification with Opposite Factors

Opposite factors (also called negatives of each other) are binomials like $(a - b)$ and $(b - a)$. These differ only by a factor of $-1$:

$b - a = -(a - b)$

When you spot opposite factors in the numerator and denominator, you can cancel them, but a factor of $-1$ remains.

Example: Simplify $\frac{4 - x}{x - 4}$

1. Recognize that $4 - x = -(x - 4)$

2. Rewrite: $\frac{-(x-4)}{x-4}$

3. Cancel $(x - 4)$: the result is $-1$

Simplifying Complex Rational Expressions

A complex rational expression (or complex fraction) has a fraction in its numerator, its denominator, or both. To simplify one, rewrite the division as multiplication by the reciprocal.

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

1. Rewrite as multiplication by flipping the bottom fraction: $\frac{x-3}{x+3} \cdot \frac{x+1}{x-1}$
2. Multiply across: $\frac{(x-3)(x+1)}{(x+3)(x-1)}$
3. Check for common factors. Here, no factors cancel, so the simplified result is: $\frac{(x-3)(x+1)}{(x+3)(x-1)}$

The restrictions are $x \neq -3$, $x \neq -1$, and $x \neq 1$ (any value that would create a zero denominator in the original expression).