Simplifying Rational Expressions Objective: Identify values excluded from the domain of a rational expression. Simplify rational expressions.

Rational Expressions The expression is an example of a rational expression. A rational expression is an algebraic fraction whose numerator and denominator are polynomials. Since division by zero is undefined, the polynomial in the denominator cannot be zero.

Example 1 State the excluded values for each rational expression. b + 7  0 a2 – a – 12  0 2x + 1  0 2x  -1 b  -7 *Use your calculator to find the zeros! x  - ½ a  -3 or 4

Example 2 Suppose a cylinder has a volume of 770 cubic inches and a diameter of 12 inches. Find the height of the cylinder. Round to the nearest tenth.

Simplify Expressions A rational expression is in simplest form when the numerator and denominator have no common factors except 1. To simplify a rational expression, divide out any common factors of the numerator and denominator.

Example 3 Which expression is equivalent to . .

Simplify Expressions You can use the same procedure to simplify a rational expression in which the numerator and denominator are polynomials.

Example 4 Simplify . State the excluded values of x. x + 4 ≠ 0

Simplify Expressions When simplifying rational expressions, look for binomials that are opposites. For example 5 – x and x – 5 are opposites because 5 – x = -(x – 5).

Example 5 Simplify . State the excluded values of x. x – 5 ≠ 0 x ≠ 5

Simplify Expressions Recall that to find the zeros of a quadratic function, you need to find the values of x when f(x) = 0. The zeros of a rational function are found the same way.

Example 6 Find the zeros of f(x) = .