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MAT 0022C/0028C Final Exam Review

MAT 0022C/0028C Final Exam Review. BY: West Campus Math Center. Topics. Factoring # 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 Problem Solving (Word Problems)

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MAT 0022C/0028C Final Exam Review

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  1. MAT 0022C/0028C Final Exam Review BY: West Campus Math Center

  2. Topics • Factoring • #1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 • Problem Solving (Word Problems) • #19, 20, 21, 22, 23, 24, 25, 26, 27, 66, 67, 68, 69, 70 • Graphing • #28, 29, 30, 31, 32, 33, 34, 35 • Exponents and Polynomials • #36, 37, 38, 39, 40, 41, 42, 43, 44, 45 • Square Roots/Radicals • #46, 47, 48, 49, 50, 51, 68 • Equations and Inequalities • #52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, • Test Taking Tips • MyMathLab Tips, How to study, General Test Taking Tips

  3. Problem #1, 2, 3 (GCF) 1) GCF = 2 2) GCF = 15m5 3) GCF = 44x2 Remember GCF for like variables = smallest exponent EXAMPLE: GCF for x5, x3 = x3 (pick smallest exponent)

  4. Factoring Trinomials (Leading Coefficient = 1) Form: x2 + bx + c Signs could be + or – in the trinomial. Form: x2 + bx + c Factored Form: (x )(x ) +c -c Sign of Bigger Factor Matches Middle Term bx } + – + Match Sign of Middle Term bx + – –

  5. Problem #4 (Factoring Trinomials) -20 Factored Form: (x )(x ) Find factors of 20 that subtractto make 8. – + Sign of bigger factor matches middle term bx 1 2 4 20 10 5 Solution: (x + 2)(x – 10)

  6. Problem #5 (Factoring Trinomials) -40 Factored Form: (x )(x ) Find factors of 40 that subtractto make 3. – Sign of bigger factor matches middle term bx + 1 2 4 5 40 20 10 8 Solution: (x + 5)(x – 8)

  7. Problem #6 (Factoring) xy + 6x – 3y – 18 GCF = x x(y + 6) GCF = -3 -3(y + 6) (x – 3) (y + 6) (y + 6)(x – 3)

  8. Problem #7 (Factoring) xy – 2yz + 7x – 14z GCF = y y(x – 2z) GCF = +7 +7(x – 2z) (x – 2z) (y + 7) (x – 2z)(y + 7)

  9. Problem #8 (Factoring) 10x2 – 8x – 15x + 12 GCF = 2x 2x(5x – 4) GCF = –3 –3(5x – 4) (5x – 4) (2x – 3) (5x – 4)(2x – 3) Live example!

  10. Factoring Trinomials (Leading Coefficient a 1) Form: ax2 + bx + c Signs could be + or – in the trinomial. 2 methods: 1) AC method 2) Guess-and-check We will illustrate both methods. +ac -ac Sign of Bigger Factor Matches Middle Term bx } + – + Match Sign of Middle Term bx + – –

  11. SIGN RULES(For all Factoring Methods) + – Sign of Bigger Factor Matches Middle Term bx } Match Sign of Middle Term bx + – + + – –

  12. Problem #9 (Factoring – AC method) -168 Find factors of 168 that subtractto make 17. Sign of bigger factor matches middle term bx 4 terms: 8x2+ 24x – 7x – 21 We now factor by grouping! 14 3 1 6 4 8 12 2 7 28 81 24 168 21 56 42 – +

  13. Problem #9 CONT… -168 4 terms: 8x2+ 24x – 7x – 21 8x2 + 24x – 7x – 21 GCF = – 7 – 7(x + 3) GCF = 8x 8x(x + 3) (x + 3) (8x – 7) –7 +24 (x + 3)(8x – 7)

  14. Problem #10 (Factoring – Guess/check method) – (4y 3)(4y 3) – (4y – 3)2 – – SAME SIGNS, Match Middle Term! 1 2y 4y 3 1y 3 4y 9 8y 16y

  15. Problem #11 (Factoring – AC method) -144 – + Find factors of 144 that subtractto make 7. Sign of bigger factor matches middle term bx 4 terms: 12x2+ 9x – 16x – 12 We now factor by grouping! 12 1 2 3 4 6 8 9 12 38 18 16 48 72 144 24 Live example!

  16. Problem #11 CONT… -144 4 terms: 12x2+ 9x – 16x – 12 12x2 + 9x – 16x – 12 GCF = – 4 – 4(4x + 3) GCF = 3x 3x(4x + 3) (4x + 3) (3x – 4) +9 –16 Live example! (4x + 3)(3x – 4)

  17. Problem #12 (Factoring) +36 Factored Form: (x )(x ) Find factors of 36 that addto make 12. SAME SIGNS, Match Middle Term! Solution: (x + 6)(x + 6) = (x + 6)2 3 4 2 1 6 6 12 18 36 9 + +

  18. Problem #13 (Factoring) – (6x 5)(6x 5) – (6x – 5)2 – – SAME SIGNS, Match Middle Term! 6x 1x 2x 1 5 4x 3x 25 12x 18x 36x 5 6x 9x

  19. Factoring Difference of Squares Form: a2– b2 Factored Form: (a + b)(a – b) where a and b are square roots. For variable, divide exponent by 2 EXAMPLES:

  20. Problem #14 (Difference of Squares) Factored Form: ( + )( – ) Solution: (2x+ 5)(2x– 5) 4x2 25 2x 5 2x 5

  21. Problem #15 (Difference of Squares) Factored Form: ( + )( – ) Solution: (5x+ 8y)(5x– 8y) 25x2 64y2 5x 8y 5x 8y

  22. Problem #16 (Difference of Squares) Factored Form: ( + )( – ) x4 625 Solution: (x2+ 25)(x2– 25) Solution: (x2 + 25)(x– 5)(x+5) x2 25 x 5 x2 25 x 5 Live example!

  23. Factoring ONLY vs. Solving Solve: x2 + 6x + 8 = 0 Factor: x2 + 6x + 8 (x )(x ) (x + 2)(x + 4) = 0 Find factors of 8 that addto make 6. +8 (x + 4) = 0 (x + 2) = 0 Same Signs, Match Middle + + 2 1 4 8 x = –4 x = –2 (x + 2)(x + 4)

  24. Problem #17 (Factoring) -60 Factored Form: (x )(x ) = 0 Find factors of 60 that subtractto make 7. Sign of bigger factor matches middle term bx – + Factored: (x – 5)(x + 12) = 0 3 6 5 2 1 4 10 30 20 15 12 60

  25. Problem #17 CONT… Factored: (x – 5)(x + 12) = 0 (x + 12) = 0 (x – 5) = 0 x = 5 x= –12

  26. Problem #18 (Factoring) First Step, make equation equal to 0 ADD 40 to both sides! Solve: x2 – 13x + 40 = 0 +40 Factored Form: (x )(x ) = 0 Find factors of 40 that addto make 13. Same Signs, Match Middle 5 4 2 1 10 8 40 20 – – Factored: (x – 5)(x – 8) = 0

  27. Problem #18 CONT… Solve: x2 – 13x + 40 = 0 Factored: (x – 5)(x – 8) = 0 (x – 8) = 0 (x – 5) = 0 x = 5 x= 8 Live example!

  28. Problem Solving (Types) • Percent and Applications • Proportions • Perimeter (Rectangle) • Pythagorean Theorem (a2 + b2 = c2)

  29. Percent Applications (2 methods) Proportion: Equation: Key Words: IS: = OF: Multiply Key Words: What number/percent  Variable What number/percent  Variable

  30. Problem #19 (Problem Solving) Proportion: 75% of 84 is what number? Solve: Equation to solve: 100x = 75 84 100x = 6300 x = 63

  31. Problem #19 (Method #2) 75% of 84is what number? Equation to solve: 0.75 84 =x Change 75%  0.75 x = 63

  32. Problem #20 (Problem Solving) Proportion: 90 is what percent of 60? Solve: Equation to solve: 60x = 90 100 60x = 9000 x = 150%

  33. Problem #20 (Method #2) 90 is what percent of 60? Equation to solve: 90 =x60 x = = 1.5 Change 1.5  150%

  34. Problem #21 (Problem Solving) Proportion: 125% of what number is 75? Solve: Equation to solve: 125x = 75 100 125x = 7500 x = 60

  35. Problem #21 (Method #2) 125% of what number is 75? Equation to solve: 1.25x= 75 Change 125%  1.25 x = = 60 Live example!

  36. Problem #22 (Problem Solving) Total = $230; Percent = 5% Part is missing (Discount Amount) Proportion: Equation to solve: 100x = 230 5 Solve: 100x = 1150 x = $11.50 (Discount) Sales Price = $230 - $11.50 = $218.50

  37. Problem #22 (Alternative Method) Total = $230 Percent = 5% Discount = 5% of 120,000 of: key word for multiply! = $11.50 (Discount) Solve: (0.05)(230) Sales Price = $230 - $11.50 = $218.50

  38. Problem #23 (Problem Solving) Total = $2000; Part = $408 Percent is missing Proportion: Equation to solve: 2000x = 408 100 Solve: 2000x = 40800 x = 20.4%

  39. Problem #23 (Alternative Method) = = 2 = 0.204 2 = 20.4% Live example!

  40. Problem #24 (Problem Solving) Just Plug-in: P = 24, W = 6 P = 2L + 2W; P = 24, W = 6 Solve: 2L + 12 = 24 24 = 2L + 2(6) Live example! L = 6 24 = 2L + 12

  41. Problem #25 (Problem Solving) Fencing goes around (Perimeter) 2 m = 48 m w w 2 m Solve: 2w + 4 = 48 w = 22 m

  42. Problem #26, 27 (Problem Solving) Proportion: 26) 7% of what number is 43.3? 27) 38.8 is 65% of what number? 27) 26)

  43. Problem #28 (Graphing) -2 -2 (-2, 0) (0, -2)

  44. Solution for #28

  45. Problem #29 (Graphing) y-intercept (Plug x = 0): 16y – 4(0) = –8 x-intercept (Plug y = 0): 16(0) – 4x = –8 Solve 16y = –8; y = Solve: – 4x = –8; x = 2 2 x-int: (2, 0) y-int: (0, )

  46. Problem #29 (Graphing) 2 (2, 0) (0, - ½)

  47. Solution for #29 Live example!

  48. Problem #30 (Graphing) Slope-intercept form: y = mx + b y = – 1 First point: (0, -1) (5, 1) Slope = (0, - 1)

  49. Problem #30 (Alternative Method) For x = 0: y = (0) – 1 y = –1 0 -1 For x = 5: y = (5) – 1 5 1 y = 1

  50. Problem #30 CONT… 0 -1 5 1 (5, 1) (0, - 1)

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