5.1, 5.2, 5.3 Properites of Exponents last revised 6/7/2014. c = Properites of Exponents. *Simplify each of the following:
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1 48 5.1, 5.2, 5.3 Properites of Exponents last revised 6/7/2014 Properites of Exponents 1. x a x b = x a+b *Simplify each of the following: a. x 4 x 8 = b. x 5 x 7 x = 2. xa xb = xa b c = d. x14 x 9 = 3. (x a ) b = x a b e. x6 y 11 z 14 x 3 y 7 z 12 = 4. x 0 = 1 f. (2x 2 y 15,000 ) 0 = g. 3x 0 = 5. x a = 1 x a h. (3x) 0 = i. x 4 x 2 = j. x3 y 4 x 2 y 8 = Negative exponents are NOT considered to be simplified. Do NOT leave them in final answers! 48
2 49 k. ( 3 5 )2 = s. ( 3x 2 y 4 6x 5 y 7) 3 = l. (3x 2 y 3 ) 4 = m. (2x 2 y 3 ) 2 ( 3xy 4 ) 2 = 6. Simplify: ( 3u 3 ) w 6 ( 2u2 v 3 w 2 ) 3 n. x 4 x 8 = o = p. 6 2 = q. 6 2 = r. 12x 4 y 3 48x 7 y 5 = Simplify each: 5x (x 2 y 3 ) 0 49
3 Scientific Notation Scientific notationis a shorthand notation for writing extremely small or large numbers. Notation: *Multiply. Write your answers in scientific notation: 6. (3.5 x ) (4.0 x ) *Write each using scientific notation: 7. (2.45 x ) (3.5 x ) 1. 9,374, trillion *Divide. Write your answers in scientific notation: x x *Write each in standard form: x x x x
4 Adding and Subtracting Polynomials monomial binomial trinomial polynomial *Add: 5x 3 +3x x 3 9x 2 +5x 3 *Find the perimeter: Vocabulary: ax n + bx n cx + d *Given: 5x 7 + 4x 6 + 3x 5 + 5x 11, find the following: a. leading coefficient 5x+3 8x+2 8x+2 6x - 4 6x - 4 b. constant term c. degree of the second term *Subtract: (3x 2 + 5x + 11) (x 2 + 7x 4) d. degree of the polynomial If a term has more than one variable, its degree is the of its exponents. *What is the degree of the expression 5x 2 y 7? *Subtract: 5x 3 2x 2 +4x 9 2x 3 +7x 2 11x+8 *Add: (3x 2 + 5x 2) + (7x 2 9x + 13) 51
5 Multiplying Polynomials 8. (4x 7)(4x + 7) *Multiply each of the following: 1. ( 3x 4 )(5x 5 ) 2. 4x(3x 7) 9. (a + b)(c + d) 3. 5a 2 b 3 c 4 (3ab 7 5ab 2 d 5 ) 10. (3x 2)(x 2 + 4x 7) 4. (x + 5)(x 3) 11. (5x + 7)(3x 2) 5. (2x 3)(3x + 5) 6. (3x 5)(2x + 4) 12. (3x 2) 2 7. (5x 1)(x + 8) 13. (3x 2 y 4 ) 2 52
6 53 Mini-Review of Sections Multiply: (x 3)(x 2 + 3x + 9) 1. Simplify: 5x 3 y 10x 4 y 5 2. Multiply: (4.3 x 10 8 ) (3.0 x ) 7. Multiply: (x 2 + 4x 2)(x 2 + x 5) 3. Divide: 0.6 x x Calculate the area of the larger rectangle in two ways. x 5 x 4. Multiply: 5x 2 y 3 z(8xyz 5 4xy 2 z 3 ) 3 5. Multiply: (3x 2 + 4)(2x 2 7) 53
7 Dividing Polynomials A. Dividing by a monomial 3. (x 3 + 4x 2 2x + 8) (x 1) Create separate fractions and then simplify each separately x4 y 3 +15x 2 y 8 10x 2 y 9 2. (8x 3 y 4x 7 y 5 + 2x 2 y 4 ) (4xy 8 ) 4. 4t3 +4t 2 9t+3 2t+3 B. Dividing by a non-monomial Use long division. Recall
8 55 5. (4x 3 + 5x 12) (2x 3) 7. (p 4 p 3 4p 2 2p 15) (p 2 + 2) 6. (x 3 27) (x 3) In Math 90, you will learn another method for dividing called Synthetic Division. This process will only work when dividing by a linear factor (those without exponents). Problems like #7 above cannot be done using synthetic division since there is an exponent in the divisor. 55
9 Introduction to Factoring Factoring is. Factoring is a process. B. Factoring by Grouping 6. x 2 (x 5) + 7(x 5) 7. 5x(x 3 + 2) 8(x 3 + 2) A. Factoring Out a Greatest Common Factor 8. 3q + 3p + qr + pr *Factor each of the following completely x w w 2 10w x 2 18x 3. 20x 5 y 3 z 2 24x 2 y 5 z 4. 14x 5 y 3 28x 7 y x 2 y c + 3ay + ac + 6y 5. 12x 3 + 4x
10 x 2 + 6x + 8x Simplify: ( 4x 3 y 6xy 5) f 2 k + 30k + 2f Review 5. Multiply: (5.6 x ) (3.0 x ) 1. Multiply: (x + 1)(x + 2)(x + 3) 6. Multiply: 4x 2 y 3 z(8xy 5 11x 2 z 3 ) 7. Multiply: (5x 2 y 8z) 2 2. Simplify: Simplify: (5x 3 y 2 ) 2 57
11 58 8. Divide: (16x 4 y x 5 y 24x 3 y 8 ) (6x 2 y 7 ) 11. Factor: 8x x x 12. Factor: 5x 2 (x + 7) 8(x + 7) 9. Divide: (x 3 + 6x 7) (x + 1) 13. Factor: 2c + ac + 3ay + 6y 10. Divide: (m 3 64) (m 4) 14. Factor: xy xz + 7y 7z 15. Factor: 4x 3 + 3x 2 y + 4xy 2 + 3y 3 58
12 Factoring Trinomials, part 1 * Factor each of the following: 7. 12p 2 96p x x x 3 y 3 19x 2 y xy 3 2. x 2 3x m m x 2 + 6x m 2 12m w 2 40w n 2 + 8n w 2 7w
13 Factoring Trinomials, part y 2 73yz + 6z 2 pattern: ax 2 + bx + c 6. 36x 2 18x 4 *Factor each of the following completely. 1. 3x x m mn 5n y 2 + 9y y 2 6y x 2 18x 9. 16x x p p c 2 5c y 3 y y 60
14 Factoring Trinomials, part 3 The a-c Method 3. 16x x x 2 + 7x x 2 12x x 2 34x 15 61
15 The Difference of Two Squares and Perfect Square Trinomials The Difference of Two Squares 9. x 8 y 8 *Factor each completely: 1. x x x x x x x 2 49y 2 5. x x x x 2 y 2 100z 2 8. x x 2 6xy + 9y
16 x 2 + 8x y The Sum & Difference of Two Cubes Memorize: a 3 + b 3 = (a + b)(a 2 ab + b 2 ) a 3 b 3 = (a b)(a 2 + ab + b 2 ) "SOAP" means... *Factor each completely: 16. w 2 10w q 2 1. x y x 2 16xy 64y x x What if you forget these formulas? 63
17 64 A General Stategy for Factoring #10. 5r Can I factor out a? 2. How many terms are there? a. if four, try. #30. 3y 2 + y + 1 b. If three, try. c. If two, try or try. #35. p 3 5p 2 4p 3. Can I factor further? The following exercises are from pages of your textbook, which are printed on the next page for your use. #45. 14u 2 11uv + 2v 2 #5. 2a #15. 3q 2 9q 12 #55. 81u 2 90uv + 25v 2 # k + k 2 #65. 12x 2 12x
18 65 65
19 Solving Equations Using the Zero Product Rule Quadratic equations are of the form 5. x 2 2x 15 = 0 Zero Product Rule: If A B = 0, Then A = 0 or B = x 2 8x + 16 = 0 Solve each of the following equations. 1. (x + 2)(x 3) = 0 7. x 2 24 = 2x 8. x 2 25 = 0 2. (x + 5)(2x 3) = x 2 50 = 0 3. (8x 5)(7x + 2) = x 3 25x = z (z 5 8 ) = x 3 50x = 0 66
20 #26 y 2 7y 8 = #48 9n 2 = #28 w 2 10w + 16 = #49 2y y 2 = 20y 6.7 #30 4x 2 11x = #35 2m 3 5m 2 12m = #62 3z(z 2) z = 3z #38 4(2x 1)(x 10)(x + 7) = #69 (x 1)(x + 2) = #46 2y 2 20y = 0 67
21 Quadratic Word Problems A. Number Problems 6.8 #16 The sum of the squares of two consecutive integers is 9 less than 10 times their sum. Find all such integers. 6.8 #10 If a number is added to two times its square, the result is 36. Find all such numbers. B. Consecutive Integer Problems Review... Consecutive Integers Consecutive Even Integers Consecutive Odd Integers The product of two consecutive odd integers is 63. Find all such integers. 6.8 #14 The product of two even consecutive integers is 48. Find all such numbers. 68
22 69 The perimeter of a rectangle is 22 cm and its area is 24cm 2. Find the length and width of this rectangle. The length of a rectangle is three times its width. Find the dimensions if the area is 48 cm 2. D. Height of a Projectile C. Area Problems 6.8 #26 A stone is dropped off a 256-ft. cliff. The height of the stone is given by h = 16t , where t is the time (in seconds). When will it hit the ground? 6.8 #18 The width of a rectangular picture frame is 2 inches less than its length. The area is 120 in 2. Find its dimensions. 69
23 70 E. The Pythagorean Theorem Review of Chapters 5 & 6 Find x: x Simplify: (5x 2 y)( 3x 4 y 3 ) 8 2. Simplify: 5x 0 3. Simplify: Simplify: (x 5 ) 3 5. Simlify: ( 3x 2 y 4 6x 5 y 2) #42 The longer leg of a right triangle is 1 cm less than twice the shorter leg. The hypotenuse is 1 cm more than twice the shorter leg. Find the length of the shorter leg. 6. Simplify: ( 4x 5 y 3 z 8x 7 y 4 z 3) 4 70
24 71 7. Write 5.37 x 10 4 in standard form. 12. Divide: (5x x 2 15x + 20) (15x 3 ) 8. Write 365,000,000 in scientific notation. 9. Multiply: (3.5 x ) (4.0 x ) 13. Simplify: (3x 2y) Divide: 6.0 x x Add: (4x + 2) + (3x 1) 15. Subtract 3x 2 4x + 8 from x 2 9x Divide: x3 +64 x Multiply: (3x + 5)(2x 7) 17. Multiply: (x 4)(x 2 + 5x 3) 71
25 The square of a number is subtracted from 60, resulting in 4. Find all such numbers. 20. The length of a rectangle is 1 ft. longer than twice its width. If the area is 78 ft 2, find the rectangle's deimensions. 19. The product of consecutive integers is 44 more than 14 times their sum. Find all such integers. 21. A right triangle has one leg that is 2 ft. longer than the other leg. The hypotenuse is 2 ft. less than twice the shorter leg. Find the lengths of all three sides of hte triangle. 72
26 Factor: x 2 + x Factor: 3a ab + 54b Factor: c Factor: 10u u Factor: y Factor: 49 + p Factor: 2x 3 + x 2 8x 4 73
27 74 Additional Review (Chapters 5&6) If Time f. ( 3m 2 n 3 ) 4 1. Write 463,000,000 in scientific notation. 2. Multiply: (2.5 x ) (6.0 x 10 4 ) g. 5x 2 y 3 10x 4 y 5 h. ( 3u 2 v w 3 ) 2 (uv 2 ) 6.0 x Divide: 8.0 x Simplify each: a. 5x 0 b. (5x) 0 5. Given 5x 2 + 3x 7 2x + 8, state find each: c. ( 2 3 )2 a. Leading coefficient: b. degree of the polynomial: d. 8 2 e. ( 8) 2 6. Find the degree of the polynomial: x 2 y 4 z + 3xy 7 z 2 2x 4 yz 3 74
28 75 7. Add: (2x 5) + (4x + 8) e. (x + 5)(x 2 x + 2) 8. Combine: (x 2 + 4x 9) (2x 2 3x + 4) + (5x 2 11) 10. Divide: 5x2 y 3 12xy 4 +10x 4 y 8x 3 y 2 9. Multiply each: a. (3x 5)(2x + 7) 11. Divide: (6x 2 5x + 4) (x 3) b. (4x + 7)(4x 7) c. (3x + 2y)(4x 9y) 12. Factor: 5x x d. (5x 8) Factor: 60xa 30xb 80ya + 40yb 75
29 Factor: x 2 3x Solve: x 2 = 8x 15. Solve: x 2 3x 28 = The length of a rectangle is 2 cm more than three times its width. Find the dimensions if the area is 56 cm Factor: 2x 2 x Factor: x Factor: x 3 27y 3 76
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