Math 025 Unit 5 Section 6.1

A fraction in which the numerator and the denominator are polynomials is called a rational expression. Examples: 3y 2 2x 3 x + 2 2x – 7 x 2 + 2x – – x 2 Remember that a fraction bar is a grouping symbol, so the last two fractions could be written as (x + 2) (2x – 7) (x 2 + 2x – 24) (16 – x 2 )

Objective: To simplify rational expressions A rational expression is in simplest form if the numerator and the denominator have no factors in common 4x 3 y 4 6x 4 y = 2x 3 y  2y 3 2x 3 y  3x  2y 3  3x = Usually when the numerator and the denominator are both monomials, it is easier to simplify by using the rules of exponents as we did in Chapter 4

Objective: To simplify rational expressions A basic rule to follow in simplifying rational expressions is that a quantity in parentheses can only cancel a quantity that is exactly like it. Simplify: x 2 – 2x – 8 x 2 – 4 ( ) = (x – 4)(x + 2) (x + 2)(x – 2) (x – 4) (x – 2) = If the fraction does not simplify originally, see if the numerator and denominator can be factored to produce some canceling.

Objective: To simplify rational expressions Simplify: 6x 5 y 12x 2 y 3 = = x 3 2y 2 Simplify: x 2 + 2x – 15 x 2 – 7x + 12 ( ) (x + 5)(x – 3) (x – 4)(x – 3) = x + 5 x – 4

Objective: To simplify rational expressions = Simplify: x 2 + 2x – – x 2 (x + 6)(x – 4) (4 – x)(x + 4) = = -(x + 6)(4 – x) (4 – x)(x + 4) - (x + 6) (x + 4) = – x + 6 x + 4

Objective: To multiply rational expressions 1. Be sure the numerators and denominators are factored. 2. Do any canceling that is possible. 3. Write the answer – usually in factored form Simplify: 10x 2 – 15x 3x – 2 12x – 8 20x – 25  5x(2x – 3) 4(3x – 2) = = x(2x – 3) 4(4x – 5)  (3x – 2) 5(4x – 5)

Objective: To multiply rational expressions 1. Be sure the numerators and denominators are factored. 2. Do any canceling that is possible. 3. Write the answer – usually in factored form Simplify: x 2 + x – 6 x 2 + 3x – 4 x 2 + 7x – x 2  = = x – 1 x + 2 – (x + 3)(x – 2) (x + 3)(x + 4)  (x + 4)(x – 1) (2 – x)(2 + x)

Objective: To divide rational expressions 1. Rewrite the problem as a multiplication problem 2. Factor the numerators and denominators. 3. Do any possible canceling Simplify: 2x 2 + 5x + 2 3x x + 4 2x 2 + 3x – 2 2x 2 + 7x – 4  = = 2x + 1 3x + 1 2x 2 + 5x + 2 2x 2 + 7x – 4 2x 2 + 3x – 2 3x x + 4 =  (2x + 1)(x + 2) (2x – 1)(x + 2)  (2x – 1)(x + 4) (3x + 1)(x + 4)

Objective: To divide rational expressions 1. Rewrite the problem as a multiplication problem 2. Factor the numerators and denominators. 3. Do any possible canceling Simplify: xy 2 – 3x 2 y 6x 2 – 2xy z 2 z 3  = = yz 2 xy 2 – 3x 2 y z 3 =  z 2 6x 2 – 2xy xy (y – 3x) z - z2z2  z3z3 2x(3x – y)