Understanding Rational Expressions

Fractions with Variables

Think back to the last time you worked with fractions, like $1/2$ or $3/4$. These are called rational numbers because they're a ratio of two integers. A rational expression is just like that, but with a twist: instead of integers, we use polynomials. A rational expression is a fraction where the numerator and the denominator are both polynomials.

A rational expression looks like $P(x)/Q(x)$ , where $P(x)$ and $Q(x)$ are polynomials, and $Q(x)$ is not the zero polynomial.

They can be simple, like $5/x$ . Or they can look more complex, like $(x^2 - 4)/(2x^2 + 3x - 1)$ . The key idea is the same: it's one polynomial divided by another.

The Domain is Key

With fractions, there’s one big rule we can never break: you can't divide by zero. The same rule applies to rational expressions. The denominator can never be equal to zero. This rule helps us define the domain of the expression, which is the set of all possible input values ( $x$ -values) that are allowed.

Finding the domain is straightforward. First, set the denominator equal to zero. Then, solve for the variable. Any values you find are the ones that must be excluded from the domain because they would make the denominator zero.

Let's find the domain for the expression $(x + 5)/(x - 3)$ .

Set the denominator to zero: $x - 3 = 0$
Solve for $x$ : $x = 3$

This means $x$ can be any real number except 3. If you tried to plug in $x = 3$ , you’d get $8/0$ , which is undefined.

How about a more complex one, like $(2x)/(x^2 - 9)$ ?

Set the denominator to zero: $x^2 - 9 = 0$
Factor the polynomial: $(x - 3)(x + 3) = 0$
Solve for $x$ : $x = 3$ and $x = -3$

For this expression, the domain is all real numbers except 3 and -3.

Simplifying Expressions

Just like you can simplify the fraction $4/8$ to $1/2$, you can often simplify rational expressions. The process involves factoring both the numerator and the denominator and then canceling out any common factors.

Let's simplify this expression:

\frac{x^2 + 5x + 6}{x^2 - 4}

First, we factor the top and bottom polynomials.

Numerator: $x^2 + 5x + 6 = (x + 2)(x + 3)$
Denominator: $x^2 - 4 = (x + 2)(x - 2)$

Now, rewrite the fraction with the factored forms:

\frac{(x+2)(x+3)}{(x+2)(x-2)}

We have a common factor of $(x+2)$ in both the numerator and denominator, so we can cancel them out. The simplified expression is:

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

An important note: The domain of the simplified expression is determined by the original denominator. In our example, the original denominator was $x^2 - 4$ , so the domain excludes $x = 2$ and $x = -2$ , even though the $(x+2)$ factor was canceled.

Operations with Rational Expressions

You can add, subtract, multiply, and divide rational expressions using the same rules you learned for regular fractions.

Multiplication and Division

Multiplying is the easiest. Just multiply the numerators together and the denominators together. It helps to factor everything first so you can cancel common factors before you multiply.

\frac{x+1}{x-2} \cdot \frac{x-2}{x+3} = \frac{(x+1)(x-2)}{(x-2)(x+3)} = \frac{x+1}{x+3}

Division is just like multiplying by the reciprocal (the "keep, change, flip" method).

\frac{x}{x-5} \div \frac{x^2}{x+1} = \frac{x}{x-5} \cdot \frac{x+1}{x^2} = \frac{x(x+1)}{x^2(x-5)} = \frac{x+1}{x(x-5)}

Addition and Subtraction

To add or subtract, you need a common denominator, just like with numerical fractions. Let's add these two expressions:

\frac{3}{x+1} + \frac{5}{x-2}

The least common denominator (LCD) is $(x+1)(x-2)$ . We need to rewrite each fraction with this denominator.

\frac{3(x-2)}{(x+1)(x-2)} + \frac{5(x+1)}{(x+1)(x-2)}

Now that they have the same denominator, we can add the numerators:

\frac{3(x-2) + 5(x+1)}{(x+1)(x-2)} = \frac{3x - 6 + 5x + 5}{(x+1)(x-2)} = \frac{8x - 1}{(x+1)(x-2)}

Mastering these basics will set you up for success as we move on to solving more complex problems involving rational expressions.

Mastering Rational Inequalities