3.8 Simplifying Rational Expressions p

Vocabulary Rational Expression: ratio of 2 polynomials Rational Expression: ratio of 2 polynomials Excluded Value: once factored (before simplified), the value that would make the denominator 0 Excluded Value: once factored (before simplified), the value that would make the denominator 0 Simplest Form: when numerator & denominator have no common factors Simplest Form: when numerator & denominator have no common factors

Excluded Value Set denominator = 0 (you know it can’t = 0) Set denominator = 0 (you know it can’t = 0) Solve for the variable Solve for the variable That is your excluded value. That is your excluded value.

EXAMPLE 1 Find excluded values Find the excluded values, if any, of the expression. a. x x SOLUTION a. The expression x + 8 is undefined when 10x = 0, or x = 0. 10x ANSWER The excluded value is 0.

EXAMPLE 1 Find excluded values Find the excluded values, if any, of the expression. b. 2y SOLUTION The expression 5 is undefined when 2y + 14 = 0, or y = – 7. 2y + 14 ANSWER The excluded value is – 7.

EXAMPLE 1 Find excluded values Find the excluded values, if any, of the expression. c. v 2 – 9 4v4v SOLUTION c.c. The expression 4v is undefined when v 2 – 9 = 0, or (v + 3)(v – 3) = 0. The solutions of the equation are – 3 and 3. v2 – 9v2 – 9 The excluded values are – 3 and 3. ANSWER

GUIDED PRACTICE for Example 1 Find the excluded values, if any, of the expression. x + 2 3x – 5 1. ANSWER The excluded value is ANSWER The excluded value is and – n – 6 2n 2 – 5n – m m 2 – 4 ANSWER The excluded value is 2, and 2. –

To simplify Factor numerator and denominator Factor numerator and denominator Cancel any common factors (anything over itself = 1) Cancel any common factors (anything over itself = 1)

EXAMPLE 2 Simplify expressions by dividing out monomials Simplify the rational expression, if possible. State the excluded values. a. r 2r SOLUTION Divide out common factor. a. r 2r2r = r 2r2r = 1 2 Simplify. ANSWER The excluded value is 0.

EXAMPLE 2 Simplify the rational expression, if possible. State the excluded values. b. 5x5x 5(x + 2) SOLUTION b. 5x5x 5(x + 2) = 5x Divide out common factor. Simplify. = x (x + 2) ANSWER The excluded value is – 2. Simplify expressions by dividing out monomials

EXAMPLE 2 Simplify the rational expression, if possible. State the excluded values. SOLUTION c. 6m 3 – 12m 2 18m 2 c. 18m 2 6m 3 – 12m 2 = 6m 2 (m – 2) 63m2m2 Factor numerator and denominator. = 6m 2 (m – 2) 6 3 m2m2 Divide out common factors. = m – 2 3 Simplify. ANSWER The excluded value is 0. Simplify expressions by dividing out monomials

EXAMPLE 2 Simplify the rational expression, if possible. State the excluded values. SOLUTION d. y 7 – y d. The expression y 7 – y is already in simplest form. ANSWER The excluded value is 7. Simplify expressions by dividing out monomials

GUIDED PRACTICE for Example a 3 22a a 3 ANSWERThe excluded value is c2c c + 5 2c2c ANSWER The excluded value is – s 2 + 8s 3s +12 ANSWER 2s2s 3 The excluded value is – x8x 8x x 2 ANSWER 1 x 2 + 2x The excluded values are 0 and – 2.

EXAMPLE 3 Simplify an expression by dividing out binomials Simplify x 2 – 3x – 10 x 2 + 6x + 8. State the excluded values. SOLUTION x 2 – 3x – 10 x 2 + 6x + 8 = (x – 5)(x + 2) (x + 4)(x + 2) Factor numerator and denominator. (x – 5)(x + 2) (x + 4)(x + 2) = = x – 5 x + 4 Divide out common factor. Simplify. ANSWER The excluded values are – 4 and – 2.

EXAMPLE 3 Simplify an expression by dividing out binomials CHECK Check your simplification using a graphing calculator. Graph y 1 x 2 – 3x – 10 x 2 + 6x + 8 and y 2 = = x – 5 x + 4 The graphs coincide. So, the expressions are equivalent for all values of x other than the excluded values (– 4 and – 2 ).

EXAMPLE 4 Recognize opposites Simplify x 2 – 7x – x 2. State the excluded values. SOLUTION x 2 – 7x – x 2 Factor numerator and denominator. (x – 3)(x – 4) – (x – 4)(4 + x) = Rewrite 4 – x as – ( x – 4). Simplify. – (x – 4)(4+ x) (x – 3)(x – 4) = Divide out common factor. –(4 + x) (x – 3) = (x – 3)(x – 4) (x – 4)(4 + x) = ANSWER The excluded values are – 4 and 4. (x + 4) (x – 3) = –

GUIDED PRACTICE for Examples 3 and 4 Simplify the rational expression. State the excluded values. 9. x 2 + 3x + 2 x 2 + 7x + 10 (x + 1) (x + 5) ANSWER The excluded values are – 2 and – y 2 – 64 y 2 – 16y + 64 ANSWER (y + 8) (y – 8) The excluded value is z – z 2 z 2 – 3z – 10 ANSWER (z + 1) (z + 2) – The excluded values are 5 and – 2.

EXAMPLE 5 Simplify a rational model 46 – 2.2x C = 100 – 18x + 2.2x 2 where x is the number of years since Rewrite the model so that it has only whole number coefficients.Then simplify the model. The average cost C (in dollars per minute) for cell phone service in the United States during the period 1991–2000 can be modeled by CELL PHONE COSTS

EXAMPLE 5 Simplify a rational model C = 46 – 2.2x 100 – 18x + 2.2x 2 Write model. = 460 – 22x 1000 – 180x + 22x 2 Multiply numerator and denominator by 10. Factor numerator and denominator. = 2(230 – 11x) 2(500 – 90x + 11x 2 ) Divide out common factor. = 2(230 – 11x) 2(500 – 90x + 11x 2 ) SOLUTION = 230 – 11x 500 – 90x + 11x 2 Simplify.

GUIDED PRACTICE for Example 5 In Example 5, approximate the average cost per minute in ANSWER The average cost per minute in 2000 is $.23/min.