1. Chapt 6. Rational Expressions, Functions, and Equations

2. 6.1 Rational Expressions and Functions  Rational Expression Polynomial divided by non-zero polynomial Polynomial divided by non-zero polynomial 120x / (100 – x) (3x 2 - 12xy – 15y 2 ) / (6x 3 – 6xy 2 ) 120x / (100 – x) (3x 2 - 12xy – 15y 2 ) / (6x 3 – 6xy 2 )  Rational Function Function defined by a rational expression Function defined by a rational expression f(x) = (120x) / (100 – x) f(x) = (120x) / (100 – x)

3. Evaluating a Function Given: f(x) = 120x / (100 – x) Given: f(x) = 120x / (100 – x) Evaluate: f(20) f(20) = 120(20) / (100 – (20)) = 2400 / 80 = 30 f(40) = 120(40) / (100 – (40)) = 4800 / 60 = 80 Evaluate: f(20) f(20) = 120(20) / (100 – (20)) = 2400 / 80 = 30 f(40) = 120(40) / (100 – (40)) = 4800 / 60 = 80

4. Domain of a Rational Function  Given: The cost (in $1000) of cleaning up a polluted lake is a function of the percentage (x) of the lake’s pollutants to be removed. It is given by the following function. f(x) = 120x / (100 – x)  What is the cost of cleaning up 50% of the pollutants? f(50) = 120(50) / (100 – 50) = 120 f(50) = 120(50) / (100 – 50) = 120

5. Domain of a Rational Function  Given the last function: f(x) = 120x / (100 – x)  What are the possible values of x?  Answer: x ≠ 100 x ≠ 100 x cannot be negative (in practical cases) x cannot be negative (in practical cases)  Domain of f: [0, 2) U (2, 100] [0, 2) U (2, 100]

6. Domain of a Rational Function  Given: f(x) = (2x + 1) / (2x 2 – x – 1)  What is the domain of f?  Solution: (2x – x – 1) (2x + 1)(x – 1) = 0 2x + 1 = 0 x – 1 = 0 x = -1/2 x = 1 Domain of f: (-∞, -1/2) U (-1/2, 1) U (1, ∞)  Solution: (2x 2 – x – 1) (2x + 1)(x – 1) = 0 2x + 1 = 0 x – 1 = 0 x = -1/2 x = 1 Domain of f: (-∞, -1/2) U (-1/2, 1) U (1, ∞) -1/2 1

7. Your Turn  Given: f(x) = (x – 5) / (2x 2 + 5x – 3)  Find the domain of f.  Solution: 2x 2 + 5x – 3 (2x - 1)(x + 3) = 0 2x – 1 = 0 x + 3 = 0 x = ½ x = -3  Domain of f: (-∞, -3) U (-3, 1/2) U (1/2, ∞)

8. Simplifying Rational Expressions  Simplify: (x 2 + 4x + 3) / (x + 1) x 2 + 4x + 3 (x + 1)(x + 3) --------------- = ------------------ = x + 1, x ≠ -1 x + 1 (x + 1) y = x + 1 y = (x 2 + 4x + 3)/(x + 1)

9. Your Turn  Simplify 1.(x 2 + 7x + 10) / (x + 2) = (x + 2)(x + 5) / (x + 2) = x + 5, x ≠ -2 = (x + 2)(x + 5) / (x + 2) = x + 5, x ≠ -2 2.(x 2 – 7x – 18) / (2x 2 + 3x – 2) = (x + 2)(x – 9) / (2x - 1)(x + 2) = (x – 9) / (2x – 1), x ≠ -2 and x ≠ 1/2 = (x + 2)(x – 9) / (2x - 1)(x + 2) = (x – 9) / (2x – 1), x ≠ -2 and x ≠ 1/2

10. Multiplying Rational Expressions  Multiply x + 4 x – 4x - 21 -------- ∙ ---------------- x – 7 x – 16 x + 4 x 2 – 4x - 21 -------- ∙ ---------------- x – 7 x 2 – 16 x + 4 (x – 7)(x + 3) = -------- · ------------------- x – 7 (x – 4)(x + 4) x + 4 (x – 7)(x + 3) = -------- · ------------------- x – 7 (x – 4)(x + 4) x + 3 = -------- x + 3 = -------- x – 4 x – 4

11. Dividing Rational Expressions  Divide (y – 25) / (2y – 2) (y + 10y +25) / (y+ 4y – 5) (y 2 – 25) / (2y – 2) (y 2 + 10y +25) / (y 2 + 4y – 5) = (y– 25) / (2y – 2) ∙ (y+ 4y – 5)/(y + 10y + 25) = (y 2 – 25) / (2y – 2) ∙ (y 2 + 4y – 5)/(y 2 + 10y + 25) (y – 5)(y + 5) (y + 5)(y – 1) (y – 5)(y + 5) (y + 5)(y – 1) = ------------------ ∙ ------------------- 2(y – 1) (y + 5)(y + 5) = ------------------ ∙ ------------------- 2(y – 1) (y + 5)(y + 5) y - 5 y - 5 = -------- 2 = -------- 2

12. Your Turn  Simplify the following x + xy 4x – 4y ----------- · ---------- x – y x x 2 + xy 4x – 4y ----------- · ---------- x 2 – y 2 x x(x + y) 4(x – y) = ------------------ · ------------ (x – y)(x + y) x x(x + y) 4(x – y) = ------------------ · ------------ (x – y)(x + y) x = 4 = 4

13. Your Turn  Simplify (y – 4) / (y + y) (y + 5y + 6) / (y – 1) (y 2 – 4) / (y 2 + y) (y 2 + 5y + 6) / (y 2 – 1) = (y – 4) / (y + y) ∙ (y – 1) / (y + 5y + 6) = (y 2 – 4) / (y 2 + y) ∙ (y 2 – 1) / (y 2 + 5y + 6) (y – 2)(y + 2) (y - 1)(y + 1) = -------------------- ∙ ------------------ y(y + 1) (y + 2)(y + 3) (y – 2)(y + 2) (y - 1)(y + 1) = -------------------- ∙ ------------------ y(y + 1) (y + 2)(y + 3) (y – 2)(y – 1) = ------------------- y(y + 3) (y – 2)(y – 1) = ------------------- y(y + 3)

14. 6.2 Adding and Subtracting Rational Expressions  Add x + 2x – 2 5x + 12 ------------------- + ------------------ x + 3x – 10 x + 3x – 10 x 2 + 2x – 2 5x + 12 ------------------- + ------------------ x 2 + 3x – 10 x 2 + 3x – 10 x + 2x – 2 + 5x + 12 x + 7x + 10 x 2 + 2x – 2 + 5x + 12 x 2 + 7x + 10 = ---------------------------- = -------------------- x + 3x – 10 x + 3x – 10 = ---------------------------- = -------------------- x 2 + 3x – 10 x 2 + 3x – 10 (x + 2) (x + 5) (x + 2) = -------------------- = ------------- (x + 5)(x – 2) (x – 2) (x + 2) (x + 5) (x + 2) = -------------------- = ------------- (x + 5)(x – 2) (x – 2)

15. Your Turn  Add x + 5x – 15 -2x + 5 ------------------- + ------------------ x + 5x + 6 x + 5x + 6 x 2 + 5x – 15 -2x + 5 ------------------- + ------------------ x 2 + 5x + 6 x 2 + 5x + 6 x + 5x – 15 - 2x + 5 x + 3x – 10 x 2 + 5x – 15 - 2x + 5 x 2 + 3x – 10 = ------------------------------ = -------------------- x + 5x + 6 x + 5x + 6 = ------------------------------ = -------------------- x 2 + 5x + 6 x 2 + 5x + 6 (x - 2) (x + 5) (x - 2) (x + 5) = -------------------- (x + 2)(x + 3) = -------------------- (x + 2)(x + 3)

16. Your Turn  Subtract 3y – 5x 4y – 6x --------------- - --------------- x – y x – y 3y 3 – 5x 3 4y 3 – 6x 3 --------------- - --------------- x 2 – y 2 x 2 – y 2 3y – 5x - (4y – 6x3y – 5x - 4y + 6x 3y 3 – 5x 3 - (4y 3 – 6x 3 ) 3y 3 – 5x 3 - 4y 3 + 6x 3 = ------------------------------- = ---------------------------- x – yx – y = ------------------------------- = ---------------------------- x 2 – y 2 x 2 – y 2 x - y x 3 - y 3 (x – y)(x 2 + xy + y 2 ) (x 2 + xy + y 2 ) = ---------------- = --------------------------- = -------------------- x – y(x – y)(x + y) (x + y) = ---------------- = --------------------------- = -------------------- x 2 – y 2 (x – y)(x + y) (x + y)

17. Finding the Least Common Denominator  Find the LCD of: 7/6x & 2/9x  Find the LCD of: 7/6x 2 & 2/9x  Solution: 1.Factor denominators 6x  2, 3, x, x 9x  3, 3, x 1.Factor denominators 6x 2  2, 3, x, x 9x  3, 3, x 2.List all factors of 1 st Denominator—2, 3, x, x 3.Add factors of 2 nd dominator not in the list —2, 3, x, x, & 3 4.LCD: product of all factors in the list—18x 4.LCD: product of all factors in the list—18x 2

18. Finding the Least Common Denominator  Find the LCD of: 7/(5x + 15x) and 9/(x + 6x + 9)  Find the LCD of: 7/(5x 2 + 15x) and 9/(x 2 + 6x + 9)  Solution: 1. Find factors in 1 st denominator 5x + 15x  5x(x + 3) 1. Find factors in 1 st denominator 5x 2 + 15x  5x(x + 3) 2. Find factors of 2 nd denominator x + 6x + 9  (x + 3)(x + 3) 2. Find factors of 2 nd denominator x 2 + 6x + 9  (x + 3)(x + 3) 3. List factors of 1 st denominator 5x(x + 3) 4. Include in the list those factors in 2 nd denominator not found in 1st 5x(x + 3)(x + 3) or 5x(x + 3) 4. Include in the list those factors in 2 nd denominator not found in 1st 5x(x + 3)(x + 3) or 5x(x + 3) 2

19. Your Turn  Find the LCD of: 1. 7 / (y – 4) and 15 / (y + 2y) 1. 7 / (y 2 – 4) and 15 / (y 2 + 2y) 1 st den: y 2 – 4 = (y + 2)(y – 2) 1 st den: y 2 – 4 = (y + 2)(y – 2) 2 nd den: y 2 + 2y = y(y + 2) 2 nd den: y 2 + 2y = y(y + 2) LCD: (y + 2)(y – 2)y LCD: (y + 2)(y – 2)y 2. 3/(y – 5y – 6) and 6/(y – 4y – 5) 2. 3/(y 2 – 5y – 6) and 6/(y 2 – 4y – 5) 1 st den: y 2 – 5y – 6 = (y – 6)(y + 1) 1 st den: y 2 – 5y – 6 = (y – 6)(y + 1) 2 nd den: y 2 – 4y – 5 = (y – 5)(y + 1) 2 nd den: y 2 – 4y – 5 = (y – 5)(y + 1) LCD: (y – 6)(y + 1)(y – 5) LCD: (y – 6)(y + 1)(y – 5)

20. 6.3 Complex Rational Expressions  Given: p =principal (amount borrowed) p =principal (amount borrowed) r = monthly interest rate r = monthly interest rate n = number of monthly payments n = number of monthly payments A = amount of month payment A = amount of month payment pr A = ----------------------- 1 1 - -------------- (1 + r) pr A = ----------------------- 1 1 - -------------- (1 + r) n  ComplexRation Expression – has complex rational expression in numerator or denominator  Complex Ration Expression – has complex rational expression in numerator or denominator

21. Simplifying Complex Rational Expression  Simplify: 1 y --- + --- x x 2 ----------- 1 x --- + --- y y 2  Find the LCD: x x y y = x 2 y 2  Multiply all terms by x 2 y 2 / x 2 y 2 = 1

22.  (x 2 y 2 )1 (x 2 y 2 )y xy 2 + y 3 ---------- + ----------- ---------------- (x 2 y 2 )x (x 2 y 2 )x 2 x 2 y 2 ----------------------------- = --------------------- (x 2 y 2 )1 (x 2 y 2 )x x 2 y + x 3 ---------- + ----------- ---------------- (x 2 y 2 )y (x 2 y 2 )y 2 x 2 y 2 xy 2 + y 3 y 2 (x + y) y 2 ------------- = -------------- = ----- x 2 y + x 3 x 2 (y + x) x 2

23. Your Turn Simplify the following: 1.((x/y) – 1) / ((x 2 /y 2 ) – 1)) Solution: (xy – y2) / (x 2 – y 2 ) = y / (x + y) Solution: (xy – y2) / (x 2 – y 2 ) = y / (x + y) 2.(1/(x + h) – 1/x) / h Solution: -1/(x(x + h)) Solution: -1/(x(x + h))

24. Skip  6.4 Division of Polynomial Expressions  6.5 Synthetic Division

25. 6.6 Rational Equations  Given: Cost (in $1000) of cleaning a lake 120x f(x) = ---------- 100 – x where x = % of pollutants to be eliminated Cost (in $1000) of cleaning a lake 120x f(x) = ---------- 100 – x where x = % of pollutants to be eliminated  Question: If $80,000 is appropriated for the cleanup, what % of pollutants can be eliminated? If $80,000 is appropriated for the cleanup, what % of pollutants can be eliminated?

26.  120x f(x) = ----------- 100 – x  Solution: 200x 80 = ----------- 100 – x 80(100 – x) = 200x 8000 – 80x = 200x 8000 = 280x x = 28.6(%) 200x 80 = ----------- 100 – x 80(100 – x) = 200x 8000 – 80x = 200x 8000 = 280x x = 28.6(%)

27. Solving Rational Equation  Solve: x + 6 x + 24 -------- + ---------- = 2 2x 5x  Note: x ≠ 0 x + 6 x + 24 10x -------- + ---------- = 10x 2 2x 5x 5(x + 6) + 2(x + 24) = 20x 5x + 30 + 2x + 48 = 20x 78 = 13x x = 6

28. Check  Solve: x + 6 x + 24 -------- + ---------- = 2 2x 5x  Note: x ≠ 0 6 + 6 6 + 24 ? ------- + ---------- = 2 2(6) 5(6) 12 30 ------- + ------- = 2 12 30

29. Solving Rational Equation (2)  Solve: x 3 -------- = ---------- + 9 x – 3 x – 3  Note: x ≠ 3 x 3 (x – 3) -------- = (x – 3) --------- + 9 x - 3 x - 3 x = 3 + (x – 3)9 x = 3 + 9x – 27 x = -24 + 9x 24 = 8x x = 3 But x cannot be 3. Thus, no solution.

30. Solving Rational Equation (3)  Solve x 9 ---- + ----- = 4 3 x  Note: x ≠ 0 x 9 (3x) ----- + ---- = (3x) 4 3 x x(x) + 3(9) = 12x x 2 + 27 = 12x x 2 – 12x + 27 = 0 (x – 3)(x – 9) = 0 x = 3, x = 9

31. Check  Solve x 9 ---- + ----- = 4 x = 3, x = 9 3 x  Note: x ≠ 0 3 9 ? 9 9 ? --- + --- = 4 ---- + ---- = 4 3 3 3 9 1 + 3 = 4 3 + 1 = 4

32. Your Turn  Solve: x + 4 x + 20 -------- + ---------- = 3 2x 3x x + 4 x + 20 -------- + ---------- = 3 2x 3x  Solution: x ≠ 0 x + 4 x + 20 6x -------- + ---------- = 6x 3 2x 3x x ≠ 0 x + 4 x + 20 6x -------- + ---------- = 6x 3 2x 3x 3(x + 4) + 2(x + 20) = 18x 3x + 12 + 2x + 40 = 18x 52 = 13x x = 4 3(x + 4) + 2(x + 20) = 18x 3x + 12 + 2x + 40 = 18x 52 = 13x x = 4

33. Your Turn  Solve: 2x 6 -28 -------- + --------- = ------------ x – 3 x + 3 x 2 - 9 2x 6 -28 -------- + --------- = ------------ x – 3 x + 3 x 2 - 9  Solution: x ≠ 3, x ≠ -3 2x 6 -28 (x – 3)(x + 3) ---------- + ---------- = (x – 3)(x + 3) ----------- (x – 3) (x + 3) x 2 - 9 x ≠ 3, x ≠ -3 2x 6 -28 (x – 3)(x + 3) ---------- + ---------- = (x – 3)(x + 3) ----------- (x – 3) (x + 3) x 2 - 9 (x + 3)2x + (x – 3)6 = -28 2x 2 + 6x + 6x – 18 = -28 2x 2 + 12x + 10 = 0 (2x + 2)(x + 5) = 0 x = -1, x = -5 (x + 3)2x + (x – 3)6 = -28 2x 2 + 6x + 6x – 18 = -28 2x 2 + 12x + 10 = 0 (2x + 2)(x + 5) = 0 x = -1, x = -5

34. 6.7 Applications (1) Rate of Work  Suppose: Tom can complete a Web site in 15 hours, while her friend Amy can complete it in 10 hours. Working together, how many hours will it take to complete one job?  Solution: Hours working together: x Hours working together: x Hour with Tom alone: 15 Hour with Tom alone: 15 Hours with Amy alone: 10 Hours with Amy alone: 10 Tom’s rate: 1/15 per hour Tom’s rate: 1/15 per hour Amy’s rate: 1/10 per hour Amy’s rate: 1/10 per hour

35.  Find an equation Rate x Time = 1 job Rate x Time = 1 job 1 1 x ---- + ---- = 1 15 10 1 1 1 1 x ---- + ---- = 1 15 10 1 1 (30) x ---- + ------ = (30) · 1 15 10 2x + 3x = 30 5x = 30 (30) x ---- + ------ = (30) · 1 15 10 2x + 3x = 30 5x = 30 x = 6 (hours) x = 6 (hours)

36. Application (2) Speed  You commute to work a distance of 40 miles and return on the same route at the end of the day. Your average rate on the return trip is 30 miles per hour faster than your average rate on the outgoing trip. If the round trip takes 2 hours, what is your average rate on the outgoing trip to work?

37.  Solution Average speed going (mph): x Average speed going (mph): x Average speed returning: x + 30 Average speed returning: x + 30  Find Equation distance = speed x time distance = speed x time time = distance / speed time = distance / speed (time going) + (time returning) = 2 (time going) + (time returning) = 2 40/x + 40/(x + 30) = 2 40/x + 40/(x + 30) = 2 (x + 30)40 + 40x = 2x(x + 30) (x + 30)40 + 40x = 2x(x + 30) 40x + 1200 + 40x = 2x 2 + 60x 40x + 1200 + 40x = 2x 2 + 60x 0 = 2x 2 - 20x – 1200 0 = 2x 2 - 20x – 1200 0 = x 2 - 10x – 600 0 = x 2 - 10x – 600 0 = (x – 30)(x + 20) 0 = (x – 30)(x + 20) x = 30; x = 30; x = -20 (has no interpretation) x = -20 (has no interpretation)

38. Applications (3) Average Cost  Cost of running a manufacturing business is described by the cost function: C(x) = (fixed cost) + cx, where x is the number of units produced. Average cost for producing one unit is described by the average function: (fixed cost) + cx) A(x) = ------------------------ x

39.  Suppose a company manufactures robots with a fixed cost of $1,000,000 and $5000 per robot. C(x) = 1,000,000 + 5000x 1,000,000 + 5000x A(x) = --------------------------- x  How many robots need to be produced to bring the average cost down to $5500?

40.  1,000,000 + 5000x A(x) = ----------------------------------- x 5500 = (1,000,000 + 5000x) / x 5500x = 1,000,000 + 5000x 500x = 1,000,000 x = 2000 ($)