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Bell Ringer

Bell Ringer. Please complete the Prerequisite Skills on Page 548 #4-12. Chapter 8: Rational Functions. Big ideas: Performing operations with rational expressions Solving rational equations. Lesson 1: Model Inverse & Joint Variation. Essential question.

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Bell Ringer

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  1. Bell Ringer Please complete the Prerequisite Skills on Page 548 #4-12

  2. Chapter 8:Rational Functions Big ideas: Performing operations with rational expressions Solving rational equations

  3. Lesson 1: Model Inverse & Joint Variation

  4. Essential question What are the differences between direct, inverse and joint variation?

  5. VOCABULARY • Inverse variation ~ The relationship of two variables x and y if there is a nonzero number a such that y = • Constant of variation ~ The nonzero constant a in a direct variation equation y=ax, an inverse variation equation y=, or a joint variation equation z=axy • Joint variation ~ A relationship that occurs when a quantity varies directly with the product of two or more other quantities.

  6. y = y c. = x 4 7 x EXAMPLE 1 Classify direct and inverse variation Tell whether xand yshow direct variation, inverse variation, or neither. Type of Variation Given Equation Rewritten Equation a.xy = 7 Inverse b.y = x + 3 Neither Direct y = 4x

  7. y= 7= ANSWER 28 The inverse variation equation is y = x 28 = –14. Whenx = –2, y = a a –2 4 x EXAMPLE 2 Write an inverse variation equation The variables xand yvary inversely, and y = 7 when x=4. Write an equation that relates xand y. Then find ywhen x = –2 . Write general equation for inverse variation. Substitute 7 for yand 4 for x. 28 = a Solve for a.

  8. Essential question What are the differences between direct, inverse and joint variation? ~ y varies directly with x if y=ax for nonzero constant a ~ y varies inversely with x if xy = a for a nonzero constant a ~ z varies jointly with x and y if z = axy for nonzero constant a

  9. Bell Ringer Y varied directly with x. If y = 36 when x = 8, find y when x = 5

  10. Lesson 4: Multiply & Divide Rational expressions

  11. Essential question What are the steps for multiplying & dividing rational expressions?

  12. VOCABULARY • Simplified form of a rational expression ~ A rational expression in which the numerator and denominator have no common factors other than 1 and -1 • Reciprocal ~ The multiplicative inverse of any nonzero number

  13. x2 – 2x – 15 x2 – 2x – 15 Simplify : x2– 9 x2– 9 (x +3)(x –5) (x +3)(x –5) = (x +3)(x –3) (x +3)(x –3) = x – 5 x – 5 = x – 3 x – 3 ANSWER EXAMPLE 1 Simplify a rational expression SOLUTION Factor numerator and denominator. Divide out common factor. Simplified form

  14. 56x7y4 7x4y3 8x 3y 8xy3 = 4y 2x y2 8 7 x x6y3y = 7x6y = 8 x y3 ANSWER The correct answer is B. EXAMPLE 3 Standardized Test Practice SOLUTION Multiply numerators and denominators. Factor and divide out common factors. Simplified form

  15. 3x –3x2 x2 + x – 20 x2 + x – 20 x2 + 4x – 5 3x 3x 3x –3x2 3x(1– x) x2 + 4x – 5 (x –1)(x +5) (x + 5)(x – 4) = 3x 3x(1– x)(x + 5)(x – 4) = (x –1)(x + 5)(3x) 3x(–1)(x – 1)(x + 5)(x – 4) = (x – 1)(x + 5)(3x) 3x(–1)(x – 1)(x + 5)(x – 4) = (x – 1)(x + 5)(3x) EXAMPLE 4 Multiply rational expressions Multiply: SOLUTION Factor numerators and denominators. Multiply numerators and denominators. Rewrite 1– xas (– 1)(x – 1). Divide out common factors.

  16. ANSWER –x + 4 EXAMPLE 4 Multiply rational expressions = (–1)(x – 4) Simplify. = –x + 4 Multiply.

  17. x + 2 x + 2 x + 2 Multiply: (x2 + 3x + 9) (x2 + 3x + 9) x3 – 27 x3 – 27 x3 – 27 x2 + 3x + 9 = 1 (x + 2)(x2 + 3x + 9) (x + 2)(x2 + 3x + 9) = = (x – 3)(x2 + 3x + 9) (x – 3)(x2 + 3x + 9) x + 2 x + 2 = x – 3 x – 3 ANSWER EXAMPLE 5 Multiply a rational expression by a polynomial SOLUTION Write polynomial as a rational expression. Factor denominator. Divide out common factors. Simplified form

  18. 6xy2 3x5 y2 8. 8xy 9x3y 18x6y4 6xy2 3x5 y2 = 72x4y2 2xy 9x3y 18 x4y2x2y2 = 18 4 x4 y2 x2y2 = 4 for Examples 3, 4 and 5 GUIDED PRACTICE Multiply the expressions. Simplify the result. SOLUTION Multiply numerators and denominators. Factor and divide out common factors. Simplified form

  19. 2x(x –5) (x –5)(x +5) 2x2 – 10x x + 3 x2– 25 2x2 x + 3 = (x) 2x 2x(x –5) (x + 3) = (x –5)(x + 5)2x (x) 2x(x –5) (x + 3) = (x –5)(x + 5)2x (x) x + 3 = x(x + 5) for Examples 3, 4 and 5 GUIDED PRACTICE 2x2 – 10x x + 3 9. x2– 25 2x2 SOLUTION Factor numerators and denominators. Multiply numerators and denominators. Divide out common factors. Simplified form

  20. x + 5 x2+x + 1 x3– 1 x + 5 x2+x + 1 = (x – 1) (x2+x + 1) 1 (x + 5) (x2+x + 1) = (x – 1) (x2+x + 1) (x + 5) (x2+x + 1) = (x – 1) (x2+x + 1) x + 5 = x – 1 for Examples 3, 4 and 5 GUIDED PRACTICE x + 5 10. x2+x + 1 x3– 1 SOLUTION Factor denominators. Multiply numerators and denominators. Divide out common factors. Simplified form

  21. Divide : 7x 7x x2 – 6x x2 – 6x 2x – 10 2x – 10 x2 – 11x + 30 x2 – 11x + 30 7x x2 – 11x + 30 = 2x – 10 x2 – 6x (x – 5)(x – 6) 7x = 2(x – 5) x(x – 6) 7x(x – 5)(x – 6) = 2(x – 5)(x)(x – 6) = 7 7 ANSWER 2 2 EXAMPLE 6 Divide rational expressions SOLUTION Multiply by reciprocal. Factor. Divide out common factors. Simplified form

  22. 6x2 + x – 15 Divide : (3x2 + 5x) 4x2 6x2 + x – 15 (3x2 + 5x) 4x2 6x2 + x – 15 1 = 3x2 + 5x 4x2 (3x + 5)(2x – 3) 1 = x(3x + 5) 4x2 (3x + 5)(2x – 3) = 4x2(x)(3x + 5) 2x – 3 2x – 3 = 4x3 4x3 ANSWER EXAMPLE 7 Divide a rational expression by a polynomial SOLUTION Multiply by reciprocal. Factor. Divide out common factors. Simplified form

  23. 11. x2 – 2x x2 – 2x 4x 4x 5x – 20 5x – 20 x2 – 6x + 8 x2 – 6x + 8 x2 – 6x + 8 4x = 5x – 20 x2 – 2x 4(x)(x – 4)(x – 2) = 5(x – 4)(x)(x – 2) 4(x)(x – 4)(x – 2) = 5(x – 4)(x)(x – 2) 4 = 5 for Examples 6 and 7 GUIDED PRACTICE Divide the expressions. Simplify the result. SOLUTION Multiply by reciprocal. Factor. Divide out common factors. Simplified form

  24. 2x2 + 3x – 5 12. (2x2 + 5x) 6x 2x2 + 3x – 5 (2x2 + 5x) 6x 2x2 + 3x – 5 1 = 6x (2x2 + 5x) (2x + 5)(x – 1) = 6x(x)(2 x + 5) (2x + 5)(x – 1) = 6x(x)(2 x + 5) x – 1 = 6x2 for Examples 6 and 7 GUIDED PRACTICE SOLUTION Multiply by reciprocal. Factor. Divide out common factors. Simplified form

  25. Essential question What are the steps for multiplying & dividing rational expressions? Multiply: multiply the numerators / multiply the denominators then simplify Divide: multiply the first expression by the reciprocal of the second expression, then follow the rules for multiplication

  26. Bell Ringer Find the least common multiple of 20 and 45.

  27. Lesson 5: Add & Subtract Rational Expressions

  28. Essential question What are the steps for adding or subtracting rational expressions with different denominators?

  29. VOCABULARY • Complex fraction ~ A fraction that contains a fraction in its numerator or denominator.

  30. 2x 5 b. – x + 6 x + 6 7 + 3 10 5 7 7 3 3 = = = a. a. 4x + + 4x 2x 4x 4x 4x 4x 2x 5 2x – 5 b. – = x + 6 x + 6 x + 6 EXAMPLE 1 Add or subtract with like denominators Perform the indicated operation. SOLUTION Add numerators and simplify result. Subtract numerators.

  31. b. a. c. + – + = = = = = = = = = 2 + 1 4x–x 7 – 5 2 4x 7 5 1 x 3x 2 3 1 1 x–2 12x 3x2 3x2 12x x–2 x2 6x 3x2 x–2 12x 3x2 12x x–2 3x 3x – 2 4x 2 d. 2x2+2 2(x2+1) + = = x2+1 x2+1 x2+1 x2+1 for Example 1 GUIDED PRACTICE Perform the indicated operation and simplify. Subtract numerators and simplify results . Add numerators and simplify results. Subtract numerators. Factor numerators and simplify results . = 2

  32. EXAMPLE 2 Find a least common multiple (LCM) Find the least common multiple of 4x2 –16 and 6x2 –24x + 24. SOLUTION STEP 1 Factor each polynomial. Write numerical factors asproducts of primes. 4x2 – 16 = 4(x2 – 4) = (22)(x + 2)(x – 2) 6x2 – 24x + 24 = 6(x2 – 4x + 4) = (2)(3)(x – 2)2

  33. EXAMPLE 2 Find a least common multiple (LCM) STEP 2 Form the LCM by writing each factor to the highest power it occurs in either polynomial. LCM = (22)(3)(x + 2)(x – 2)2 = 12(x + 2)(x – 2)2

  34. x + 1 7 x x + 1 + 9x2 3x2 + 3x 7 7 x x = + + 9x2 9x2 3x2 + 3x 3x(x + 1) 7 3x x 9x2 3x + 3x(x + 1) EXAMPLE 3 Add with unlike denominators Add: SOLUTION To find the LCD, factor each denominator and write each factor to the highest power it occurs. Note that9x2 = 32x2and3x2 + 3x = 3x(x + 1), so the LCD is 32x2 (x + 1) = 9x2(x 1 1). Factor second denominator. LCD is 9x2(x + 1).

  35. 3x2 7x + 7 = + 9x2(x + 1) 9x2(x + 1) 3x2+ 7x + 7 = 9x2(x + 1) EXAMPLE 3 Add with unlike denominators Multiply. Add numerators.

  36. x – 3 x + 2 x + 2 –2x –1 –2x –1 – x– 3 2x – 2 2x – 2 x2 – 4x + 3 x2 – 4x + 3 – x + 2 x + 2 – 2x – 1 – 2x – 1 – = (x – 1)(x – 3) (x – 1)(x – 3) 2(x – 1) 2(x – 1) – = 2 x2 – x – 6 – 4x – 2 – 2 = 2(x – 1)(x – 3) 2(x – 1)(x – 3) EXAMPLE 4 Subtract with unlike denominators Subtract: SOLUTION Factor denominators. LCD is 2(x  1)(x  3). Multiply.

  37. x2 – x – 6 – (– 4x – 2) = 2(x – 1)(x – 3) x2+ 3x – 4 = 2(x – 1)(x – 3) (x –1)(x + 4) = 2(x – 1)(x – 3) x + 4 = 2(x –3) EXAMPLE 4 Subtract with unlike denominators Subtract numerators. Simplify numerator. Factor numerator. Divide out common factor. Simplify.

  38. STEP 1 Factor each polynomial. Write numerical factors asproducts of primes. STEP 2 Form the LCM by writing each factor to the highest power it occurs in either polynomial. for Examples 2, 3 and 4 GUIDED PRACTICE Find the least common multiple of the polynomials. 5. 5x3and 10x2–15x 5x3 = 5(x) (x2) 10x2 – 15x= 5(x) (2x – 3) LCM = 5x3 (2x – 3)

  39. STEP 1 Factor each polynomial. Write numerical factors asproducts of primes. 23(x – 2) 8x – 16 = 8(x – 2) = 4 3(x – 2 )(x + 3) 12x2 + 12x – 72 = 12(x2 + x – 6) = STEP 2 Form the LCM by writing each factor to the highest power it occurs in either polynomial. LCM = 8 3(x – 2)(x + 3) for Examples 2, 3 and 4 GUIDED PRACTICE Find the least common multiple of the polynomials. 6. 8x – 16 and 12x2 + 12x – 72 = 24(x – 2)(x + 3)

  40. 3 3 1 1 7. – – 4x 4x 7 7 3 1 4x 7 – 7 4x 4x 7 – 4x 21 7(4x) 4x(7) = 21 – 4x 28x for Examples 2, 3 and 4 GUIDED PRACTICE SOLUTION LCDis 28x Multiply Simplify

  41. x x 1 1 8. + + 3x2 3x2 9x2 – 12x 9x2 – 12x 1 = + x 3x2 3x(3x – 4) 1 = + x 3x – 4 3x2 3x(3x – 4) 3x – 4 3x – 4 x + x2 x 3x2 (3x – 4) = 3x2 (3x – 4) for Examples 2, 3 and 4 GUIDED PRACTICE SOLUTION Factor denominators LCD is 3x2 (3x – 4) Multiply

  42. 3x – 4 + x2 3x2 (3x – 4) x2 + 3x – 4 3x2 (3x – 4) for Examples 2, 3 and 4 GUIDED PRACTICE Add numerators Simplify

  43. x x 5 5 9. + + x2 – x – 12 x2 – x – 12 12x – 48 12x – 48 x = + 5 (x+3)(x – 4) 12 (x – 4) x + 3 5 = + 12 x + 3 x 12(x – 4) (x + 3)(x – 4) 12 5(x + 3) 12x = + 12(x + 3)(x – 4) 12(x + 3)(x – 4) for Examples 2, 3 and 4 GUIDED PRACTICE SOLUTION Factor denominators LCD is 12(x – 4) (x+3) Multiply

  44. 12x + 5x + 15 = 12(x + 3)(x – 4) = 17x + 15 12(x +3)(x + 4) for Examples 2, 3 and 4 GUIDED PRACTICE Add numerators Simplify

  45. x + 1 x + 1 6 6 10. – – 6 x + 1 – x2 + 4x + 4 x2 + 4x + 4 x2 – 4 x2 – 4 = (x – 2)(x + 2) x + 2 x – 2 (x+ 2)(x + 2) x + 2 x – 2 6 – = (x – 2)(x + 2) x + 1 (x + 2)(x + 2) 6x + 12 x2 – 2x + x – 2 (x – 2)(x + 2)(x + 2) – = (x + 2)(x + 2)(x – 2) for Examples 2, 3 and 4 GUIDED PRACTICE SOLUTION Factor denominators LCD is (x – 2) (x+2)2 Multiply

  46. = x2 – 2x + x – 2 – (6x + 12) = (x + 2)2(x – 4) x2 – 7x –14 (x + 2)2 (x – 2) for Example 2, 3 and 4 GUIDED PRACTICE Subtract numerators Simplify

  47. 5x 3x + 8 5 5 5 x + 4 x + 4 x + 4 1 1 1 2 2 2 + + + x + 4 x + 4 x + 4 x x x = x(x+4) x(x+4) 5x = x + 2(x + 4) = EXAMPLE 6 Simplify a complex fraction (Method 2) Simplify: SOLUTION The LCD of all the fractions in the numerator and denominator is x(x + 4). Multiply numerator and denominator by the LCD. Simplify. Simplify.

  48. – – – 11. 7 7 7 – – – 10 10 10 = – 5x = 3 (2x – 7) 30 30 x x x x x x x x x 6 6 3 5 6 3 5 3 5 for Examples 5 and 6 GUIDED PRACTICE Multiply numberator and denominator by theLCD Simplify

  49. 4 4 4 – – – 12. + + + 3 3 3 = = x x 2 – 4x 2 + 3x 2 (1 – 2x ) 2 2 2 2 2 2 = x x x x x x 2 + 3x for Examples 5 and 6 GUIDED PRACTICE Multiply numberator and denominator by theLCD Simplify Simplify

  50. 3 3 3 x + 5 x + 5 x + 5 13. 2 2 2 1 1 1 + + + x – 3 x – 3 x – 3 x + 5 x + 5 x + 5 = = 3x – 3 3x + 7 3(x – 3) = 3x + 7 (x + 5)(x – 3) (x + 5)(x – 3) for Examples 5 and 6 GUIDED PRACTICE Multiply numberator and denominator by theLCD Simplify Simplify

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