Chapter P: Review Exercises Exercises 73-74 In Exercises 73-74, perform the indicated operations. Write the Exercises 1-7 resulting polynomial in standard form and indicate its degree. In Exercises 1-2, evaluate each algebraic expression for the given 73. (-ar' + 7x - 9x + 3) + (14r) + 3p - 11x - 7) value or values of the variable(s). 74. (13x4 - Br) + 2x]) - (5x* - 3x3 + 2x] - 6) 1. 3 + 6(x - 2)' for x = 4 2 x - 5(x - y) for x = 6 and y = 2 Exercises 84-85 3. You are riding along an expressway traveling & miles per In Exercises 84 88, find each product. hour. The formula 84. (x + 7y)(3x - 5y) 85. (3x - 5y) 5 = 0.015r + x + 10 models the recommended safe distance, 3, in feet, between Exercises 89-90, 93-96 your car and other cars on the expressway. What is the In Exercises 89-105, factor completely, or state that the recommended safe distance when your speed is 60 miles per polynomial is prime hour? 89. 15x3 + 3x2 90. x - 1lr + 28 In Exercises 4 7 ler A = (a, b. c). B = [a, c. d, el. and 93. x3 + 16 94. 3r - 9r - 30r C = [a. d, f. g)- Find the indicated set. 95. 207 - 361 96. x3 - 1x - 9x + 27 4. AnB 5. AUB 6. AUC 7. COA Exercises 109-110 In Exercises 109-111, simplify each rational expression. Also, list Exercises 19-22 all members that must be excluded from the domain. In Exercises 19-22, simplify each algebraic expression. 109. x' + 21 110. * + 3x - 18 19. 5(2r - 3) + 7x x+2 1 - 36 20. 4(5x) + [(3y) + (-3y)] - (-x) Exercises 112-113 21. 3(4y - 5) - (7y + 2) 21. 8 - 213 - (5x - 1)] In Exercises 112-114, multiply or divide as indicated. 112. & + 6r + 9 x+ 3 113. fox + 2 31 + x Exercises 24-27 1 - 4 x - 2 x- - 1 *-1 Evaluate each exponential expression in Exercises 24-27 24. (-3)(-2) 25. 24+ 41 26. 53.5 27. 3 Exercises 41-44 Use the product rule to simplify the expressions in Exercises 41-44. In Exercises 43-44, assume that variables represent nonnegative real numbers. 41. V 300 42. V 1212 43. V 10x - V Zx 44. VP\f2. Evaluate z 5(x y) for x = 6 and T 62 5(6 2) 36 5(4) = 36 20 16 3. Using the formula S = 0.01522 + = + 10, find the recommended safe distance when = 60: S = 0.015(60)% + 60 + 10 = 0.015 x 3600 + 60 + 10 = 54 + 60 + 10 = 124 Exercises 4-7 Given sets: A = {a, b, c}, B = {a, c, d, e}, C = {a, d, f, g} 4. Find An B: AnB = {a, c} 5. Find A U B: AUB = {a, b, c, d, e } 6. Find A UC: AUC = {a, b, c, d, f, g} 7. Find C n A: CnA = {a}Exercises 19-22 19. Simplify 5 (2x - 3) + 7x: 5(2x - 3) + 7x = 10x - 15 + 7x = 17x - 15\f\f\f\f\fExercise 73: (-623 + 7x2 - 9x + 3) + (1423 + 3x2 - 11x - 7) Combine like terms: (-623 + 1423) + (7x2 + 3x2) + (-9x - 11x) + (3-7) 8x3 + 10x2 - 20x - 4 The resulting polynomial is 8x3 + 10x2 - 20x - 4 with a degree of 3.Exercise 74: (13x4 - 823 + 2x2) - (524 - 323 + 2x2 - 6) Distribute the negative sign and combine like terms: 13x4 - 8x3 + 2x2 - 524 + 323 2x2 + 6 (13x4 - 524) + (-823 + 323) + (2ac2 - 2ac2) + 6 824 - 5203 + 6 The resulting polynomial is 8x4 - 5x3 + 6 with a degree of 4.Exercise 84: (a + 7y) (3x - 5y) Use the distributive property (FOIL method): ac (3ac ) + x( -5y) + 7y(3x) + 7y(-5y) 3x2 - 5xy + 21xy - 35y2 3x2 + 16xy - 35y- The product is 3x2 + 16xy - 35y2. Exercise 85: (3x - 5y)2 Use the square of a binomial formula (a - 6) 2 = a2 - 2ab + 62: (3x)2 - 2(3x) (5y) + (-5y)2 9x2 - 30xy + 25yz The product is 9x2 - 30xy + 25y2.Exercise 89: 1523 + 3x2 Factor out the greatest common factor (GCF): 3x2 (5x + 1) The factored form is 3x2 (5x + 1). Exercise 90: x2 - 11x + 28 Find two numbers that multiply to 28 and add to -11, which are -7 and -4: (2 - 7) (20 - 4) The factored form is (a - 7) (x - 4).Exercise 93: 2 2 + 16 This is a sum of squares and cannot be factored over the real numbers. Therefore, the polynomial is prime. Exercise 94: 3x4 - 923 - 30x2 Factor out the GCF: 3ac2 (x2 - 3x - 10) Then factor the quadratic: 3ac2 ( x - 5) (2 + 2) The factored form is 3x2 (x - 5) (x + 2).Exercise 95: 20x7 - 3623 Factor out the GCF: 4ac 3 (5ac 4 - 9) The factored form is 4x3 (5x4 - 9). Exercise 96: 23 - 3x2 - 9x + 27 Factor by grouping: ac2 ( 2 - 3) - 9(2 - 3) (202 - 9) (2 - 3) (2 + 3) (2 - 3) (2 - 3) The factored form is (x + 3) (x - 3)2.Exercise 109: x+22 r+2 Factor the numerator: z*(242) T+2 Cancel the common factor: 3')2 Excluded value: = 2 Exercise 110: :52133318 x236 Factor both the numerator and the denominator: (z+6)(z3) (z46)(z6) Cancel the common factor: r3 z6 Excluded values: x # 6,6 Exercise 112: z46z+9 z+3 x24 r2 Factor the numerator and denominator: (m}S)2 - z+3 (z4+2)(z2) z2 Combine and simplify: (z+3) (z+2)(z2) Excluded values: x # 2, 2 Exercise 113: 6z+2 . 3z'4=x z?2-1 * -1 When dividing by a fraction, multiply by its reciprocal: 6x-+2 > r1 e | 3x24-x Factor the numerators and denominators: 2(3z+1) (z1)(z+1) g1 A z(3z+1) Now, cancel the common factors (z 1) and (3z + 1): 2 (z+1)-x 2 z(z+1) The simplified form is Excluded values: x # 1,0, 1. \f