Lesson 28 Simplifying rational expressions Rational expressions A

Lesson 28 Simplifying rational expressions

Rational expressions • A rational expression is the quotient of 2 polynomials, such as 2 x 2 + 4 x-4 • x+3 • with a denominator of a degree that is greater then or equal to 1. • The denominator can not equal 0. • The above rational expression is undefined when x = -3, so -3 is an excluded value from the domain.

Excluded value • In any function, an excluded value is a domain value that makes the function undefined. • Not all rational expressions have excluded values, and some expressions have more than 1 excluded value. • Examples: -3 x -6 4 x 2 + 16 • 4 x 2 + 1 • These have no excluded values

Simplifying rational expressions • A rational expression is simplified when there are no common factors, other than 1, in the numerator and denominator. • Examples: • simplified not simplified • x 3 , x+3 x 6 y 2 (x+3)(x-7) 4

Quotient of powers property (quotient rule) • For a not equal to 0, and integers m and n • am = am-n • an • Simplify: 2 m 3 x 4 4 y 5 z • m 2 x 7 8 y 2 z 5

Simplifying by dividing out common factors • Factor out common factors in numerator and denominator. • Examples: • 4 x+8 3 b 2 -27 2 x 2 -8 • 8 x+16 b+3 x+2

Simplifying by factoring out -1 • Factoring out a -1, can allow you to have common factors. • Examples: (identify excluded values) • 6 x-12 40 - 30 x • 6 -3 x 45 x-60

Simplifying rational expressions with trinomials • To find the excluded values, set the denominator equal to 0 • Example: -2 x 2 -4 x+30 = -2(x 2+2 x-15) 2(x+5)(x-3) • 3 x 2 +21 x+30 3(x 2+7 x+10) 3(x+5)(x+2) • = -2(x-3) • 3(x+2) • Excluded values are -5 and -2

example • Identify excluded values and simplify: • 4 x 2 - 10 x +6 • -2 x 2 - x = 3 • 2 x 2 - 4 x - 6 • 2 x 3 - 2 x