Rational Expressions

A rational expression is simply an algebraic fraction where both the numerator and the denominator are polynomials. Just like regular numeric fractions, you can simplify, add, subtract, multiply, and divide rational expressions.

Simplifying Rational Expressions

To simplify a rational expression, you need to factor both the numerator and the denominator completely, and then cancel out any common factors.

Example: Simplify $\frac{x^2 - 9}{x^2 - x - 6}$

1. Factor the numerator: $x^2 - 9$ is a difference of squares, which factors to $(x - 3)(x + 3)$.
2. Factor the denominator: $x^2 - x - 6$ factors to $(x - 3)(x + 2)$.
3. Rewrite and cancel: $\frac{(x - 3)(x + 3)}{(x - 3)(x + 2)}$ Cancel the common factor of $(x - 3)$: $\frac{x + 3}{x + 2}$ (Note: $x \neq 3$ and $x \neq -2$ because the original denominator cannot be zero).

Multiplying and Dividing

Multiplying rational expressions works just like multiplying regular fractions: factor everything first, multiply the numerators together, multiply the denominators together, and cancel any common factors from the top and bottom.

Dividing rational expressions requires you to flip (find the reciprocal of) the second fraction and then follow the rules for multiplication.

Adding and Subtracting

To add or subtract rational expressions, they must have a Common Denominator.

Example: Add $\frac{2}{x - 1} + \frac{3}{x + 2}$

1. Find the Least Common Denominator (LCD): The denominators are $(x - 1)$ and $(x + 2)$. Since they share no common factors, the LCD is their product: $(x - 1)(x + 2)$.
2. Rewrite each fraction with the LCD: Multiply the top and bottom of the first fraction by $(x + 2)$ and the second by $(x - 1)$: $\frac{2(x + 2)}{(x - 1)(x + 2)} + \frac{3(x - 1)}{(x - 1)(x + 2)}$
3. Combine the numerators over the common denominator: $\frac{2(x + 2) + 3(x - 1)}{(x - 1)(x + 2)}$
4. Distribute and simplify the numerator: $\frac{2x + 4 + 3x - 3}{(x - 1)(x + 2)}$ $\frac{5x + 1}{(x - 1)(x + 2)}$

Always check if your final numerator can be factored to simplify further with the denominator. In this case, $5x + 1$ cannot be factored, so you have found your final answer!