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11.2

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The document defines rational expressions as expressions that can be written as a ratio of two polynomials where the denominator is not 0. It provides examples of finding excluded values, which make the expression undefined, by setting the denominator equal to 0. It also discusses simplifying rational expressions by dividing out common factors in the numerator and denominator and putting them in simplest form. Examples are provided of factoring, dividing out monomials, recognizing opposites, and simplifying a rational model.

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11.2
Rational Expression  Rational Expression - an expression that can be written as a ratio of two polynomials where the denominator is not 0.  A rational expression is undefined when the denominator is 0.  Excluded value – number that makes a rational expression undefined. 2  Example: x 3 is undefined when x 3. So, 3 is an excluded value
Example 1 Find excluded values Find the excluded values, if any, of the expression. 5 a. x + 8 b. 10x 2y + 14 4v 7w + 2 c. 2 d. v –9 8w 2 + w + 5 SOLUTION a. The expression x + 8 is undefined when 10x = 0, or 10x x = 0. ANSWER The excluded value is 0.
Example 1 Find excluded values 5 b. The expression is undefined when 2y + 14 = 0, 2y + 14 or x = – 7. ANSWER The excluded value is – 7. 4v is undefined when 2 c. The expression v – 9 = 0, or v2 – 9 ( v + 3) ( v – 3) = 0. The solutions of the equation are – 3 and 3. ANSWER The excluded values is – 3 and 3.
Example 1 Find excluded values 7w + 2 d. The expression 2 + w + 5 is undefined when 8w 8w 2 + w + 5 = 0. The discriminant is b 2 – 4ac = 1 2 – 4 (8 ) ( 5) < 0. So, the quadratic equation has no real roots. ANSWER There are no excluded values.
Simplifying a Rational Expression 1. Factor the numerator & denominator. 2. Divide out any common factors.  A rational expression is in simplest form if the numerator and denominator have no factors in common other than 1.
Example 2 Simplify expressions by dividing out monomials Simplify the rational expression, if possible. State the excluded values. r 5x a. b. 2r 5( x + 2) c. 6m 3 – 12m 2 d. y 18m 2 7 –y SOLUTION r r a. = Divide out common factor. 2r 2r 1 = Simplify. 2
Example 2 Simplify expressions by dividing out monomials ANSWER The excluded value is 0. b. 5x 5•x = Divide out common factor. 5(x + 2) 5 • (x + 2) x = Simplify. x +2 ANSWER The excluded value is – 2. 6m 3 – 12m 2 6m 2(m – 2) c. = Factor numerator and denominator. 2 18m 6 • 3 • m2 6m 2(m – 2) = Divide out common factors. 6 • 3 • m2
Example 2 Simplify expressions by dividing out monomials m –2 = Simplify. 3 ANSWER The excluded value is 0. y d. The expression is already in simplest form. 7–y ANSWER The excluded value is 7.
Example 3 Multiple Choice Practice 6x 2 + 32x + 10 Simplify 2 – 1 to lowest terms. 9x 2(3x + 5) 2(x + 5) 3x – 1 3x – 1 2(x + 1) 2(x + 5) 9(x – 1) 3x + 1 SOLUTION 6x 2 + 32x + 10 2(x + 5) (3x + 1) Factor numerator and = 9x 2 – 1 (3x – 1) (3x + 1) denominator.
Example 3 Multiple Choice Practice 2(x + 5) (3x + 1) = Divide out common factor. (3x – 1) (3x + 1) 2(x + 5) = Simplify. 3x – 1 ANSWER The correct answer is B.
Example 4 Recognize opposites x 2 – 7x + 12 Simplify . State the excluded values. 16 – x 2 x 2 – 7x + 12 (x – 3) (x – 4) Factor numerator and = 16 – x 2 (4 – x) (4 + x) denominator. (x – 3) (x – 4) = Rewrite 4 – x as –(x – 4). –(x – 4) (4 + x) (x – 3) (x – 4) = Divide out common factor. –(x – 4) (4 + x) x–3 = Simplify. –(4 + x)
Example 4 Recognize opposites x–3 a a =– Write –b as – . x+4 b ANSWER The excluded values are –4 and 4.
Example 5 Simplify a rational model CELL PHONE COSTS 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 the rational function 46 – 2.2x C = 100 – 18x + 2.2x 2 where x is the number of years since 1991. Rewrite the model so that it has only whole number coefficients. Then simplify the model.
Example 5 Simplify a rational model SOLUTION 46 – 2.2x C = Write model. 100 – 18x + 2.2x 2 460 – 22x Multiply numerator and = denominator by 10. 1000 – 180x + 22x 2 2(230 – 11x) Factor numerator and = denominator. 2(500 – 90x + 11x 2) 2(230 – 11x) = Divide out common factor. 2(500 – 90x + 11x 2)
Example 5 Simplify a rational model 230 – 11x = Simplify. 500 – 90x + 11x 2
11.2 Warm-Up

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11.2

  • 2. Rational Expression  Rational Expression - an expression that can be written as a ratio of two polynomials where the denominator is not 0.  A rational expression is undefined when the denominator is 0.  Excluded value – number that makes a rational expression undefined. 2  Example: x 3 is undefined when x 3. So, 3 is an excluded value
  • 3. Example 1 Find excluded values Find the excluded values, if any, of the expression. 5 a. x + 8 b. 10x 2y + 14 4v 7w + 2 c. 2 d. v –9 8w 2 + w + 5 SOLUTION a. The expression x + 8 is undefined when 10x = 0, or 10x x = 0. ANSWER The excluded value is 0.
  • 4. Example 1 Find excluded values 5 b. The expression is undefined when 2y + 14 = 0, 2y + 14 or x = – 7. ANSWER The excluded value is – 7. 4v is undefined when 2 c. The expression v – 9 = 0, or v2 – 9 ( v + 3) ( v – 3) = 0. The solutions of the equation are – 3 and 3. ANSWER The excluded values is – 3 and 3.
  • 5. Example 1 Find excluded values 7w + 2 d. The expression 2 + w + 5 is undefined when 8w 8w 2 + w + 5 = 0. The discriminant is b 2 – 4ac = 1 2 – 4 (8 ) ( 5) < 0. So, the quadratic equation has no real roots. ANSWER There are no excluded values.
  • 6. Simplifying a Rational Expression 1. Factor the numerator & denominator. 2. Divide out any common factors.  A rational expression is in simplest form if the numerator and denominator have no factors in common other than 1.
  • 7. Example 2 Simplify expressions by dividing out monomials Simplify the rational expression, if possible. State the excluded values. r 5x a. b. 2r 5( x + 2) c. 6m 3 – 12m 2 d. y 18m 2 7 –y SOLUTION r r a. = Divide out common factor. 2r 2r 1 = Simplify. 2
  • 8. Example 2 Simplify expressions by dividing out monomials ANSWER The excluded value is 0. b. 5x 5•x = Divide out common factor. 5(x + 2) 5 • (x + 2) x = Simplify. x +2 ANSWER The excluded value is – 2. 6m 3 – 12m 2 6m 2(m – 2) c. = Factor numerator and denominator. 2 18m 6 • 3 • m2 6m 2(m – 2) = Divide out common factors. 6 • 3 • m2
  • 9. Example 2 Simplify expressions by dividing out monomials m –2 = Simplify. 3 ANSWER The excluded value is 0. y d. The expression is already in simplest form. 7–y ANSWER The excluded value is 7.
  • 10. Example 3 Multiple Choice Practice 6x 2 + 32x + 10 Simplify 2 – 1 to lowest terms. 9x 2(3x + 5) 2(x + 5) 3x – 1 3x – 1 2(x + 1) 2(x + 5) 9(x – 1) 3x + 1 SOLUTION 6x 2 + 32x + 10 2(x + 5) (3x + 1) Factor numerator and = 9x 2 – 1 (3x – 1) (3x + 1) denominator.
  • 11. Example 3 Multiple Choice Practice 2(x + 5) (3x + 1) = Divide out common factor. (3x – 1) (3x + 1) 2(x + 5) = Simplify. 3x – 1 ANSWER The correct answer is B.
  • 12. Example 4 Recognize opposites x 2 – 7x + 12 Simplify . State the excluded values. 16 – x 2 x 2 – 7x + 12 (x – 3) (x – 4) Factor numerator and = 16 – x 2 (4 – x) (4 + x) denominator. (x – 3) (x – 4) = Rewrite 4 – x as –(x – 4). –(x – 4) (4 + x) (x – 3) (x – 4) = Divide out common factor. –(x – 4) (4 + x) x–3 = Simplify. –(4 + x)
  • 13. Example 4 Recognize opposites x–3 a a =– Write –b as – . x+4 b ANSWER The excluded values are –4 and 4.
  • 14. Example 5 Simplify a rational model CELL PHONE COSTS 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 the rational function 46 – 2.2x C = 100 – 18x + 2.2x 2 where x is the number of years since 1991. Rewrite the model so that it has only whole number coefficients. Then simplify the model.
  • 15. Example 5 Simplify a rational model SOLUTION 46 – 2.2x C = Write model. 100 – 18x + 2.2x 2 460 – 22x Multiply numerator and = denominator by 10. 1000 – 180x + 22x 2 2(230 – 11x) Factor numerator and = denominator. 2(500 – 90x + 11x 2) 2(230 – 11x) = Divide out common factor. 2(500 – 90x + 11x 2)
  • 16. Example 5 Simplify a rational model 230 – 11x = Simplify. 500 – 90x + 11x 2