Algebra 1 Section 13.5

Mixed Rational Expression The sum of a polynomial and a rational expression is a mixed rational expression.

Example 1 Simplify 4y + . 3 y – 2 4y 1 + 3 y – 2 • y – 2 4y2 – 8y =

Definition A complex rational expression is an algebraic expression of the form n/d in which the numerator, n, or the denominator, d, or both are rational expressions.

Complex Rational Expressions Expressions containing fractions in the numerator or the denominator should always be simplified.

Example 2 Simplify . 4 x y ÷ 4 = 4 x y = x y • 1 4 = x 4y

Example 3 Simplify . x + 2 x2 x x + 2 x2 x • = 1 x x

Complex Rational Expressions There are two methods for simplifying complex rational expressions. Multiply the numerator by the reciprocal of the denominator.

Complex Rational Expressions There are two methods for simplifying complex rational expressions. Multiply the numerator and denominator by their LCD.

Example 4 Simplify . x2 – 9 4x x + 3 8x2 Method 1 • x2 – 9 4x 8x2 =

Example 4 • x2 – 9 4x 8x2 x + 3 = • (x – 3)(x + 3) 4x 4x • 2x x + 3 =

Example 4 Method 2 x2 – 9 • 8x2 4x (x2 – 9)2x = x + 3 x + 3 • 8x2 8x2

Example 5 For this example, Method 2 will be easier. We should multiply the numerator and denominator both by their LCD, b2.

Example 5 a b 1 – a2 b2 1 + b2 b2 + a2 = b2 – ab b2 = b2 + a2 b(b – a)

Homework: pp. 554-556