5-2 Dividing Polynomials. Simplify. 2. (3a 2 b 6ab + 5ab 2 )(ab) 1 SOLUTION: 4. (2a 2 4a 8) (a + 1) SOLUTION: 6. (y 5 3y 2 20) (y 2) SOLUTION:
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1 Simplify. 2. (3a 2 b 6ab + 5ab 2 )(ab) 1 4. (2a 2 4a 8) (a + 1) 6. (y 5 3y 2 20) (y 2) esolutions Manual - Powered by Cognero Page 1
2 Simplify. 8. (10x x + 20) (5x + 5) Simplify esolutions Manual - Powered by Cognero Page 2
3 ENERGY Compact fluorescent light (CFL) bulbs reduce energy waste. The amount of energy waste that is reduced each day in a certain community can be estimated by b 2 + 8b, where b is the number of bulbs. Divide by b to find the average amount of energy saved per CFL bulb. The average amount of energy saved per CFL bulb is: Simplify. 20. (a 2 8a 26) (a + 2) esolutions Manual - Powered by Cognero Page 3
4 22. (z 4 3z 3 + 2z 2 4z + 4)(z 1) (6a 2 3a + 9) (3a 2) esolutions Manual - Powered by Cognero Page 4
5 (6z 6 + 3z 4 9z 2 )(3z 6) 1 Synthetic division: esolutions Manual - Powered by Cognero Page 5
6 32. CCSS REASONING A rectangular box for a new product is designed in such a way that the three dimensions always have a particular relationship defined by the variable x. The volume of the box can be written as 6x x x + 30, and the height is always x + 2. What are the width and length of the box? Divide the function by the height (x + 2) to find the length and width. Use synthetic division. The depressed polynomial is. Since the volume of the box is the product of length, width and height, the width and length of box are (2x + 3)(3x + 5). 34. ENTERTAINMENT A magician gives these instructions to a volunteer. Choose a number and multiply it by 4. Then add the sum of your number and 15 to the product you found. Now divide by the sum of your number and 3. a. What number will the volunteer always have at the end? b. Explain the process you used to discover the answer. a. Let x be the number. b. Sample answer: Let x be the number. Multiply the x by 4 to get 4x. Then add x +15 to the product to get 5x Divide the polynomial by x + 3. The quotient is 5. esolutions Manual - Powered by Cognero Page 6
7 Simplify. 36. (x 4 y 4 ) (x y) 38. (a 3 b 2 a 2 b + 2b)( ab) esolutions Manual - Powered by Cognero Page 7
8 42. MULTIPLE REPRESENTATIONS Consider a rectangle with area 2x 2 + 7x + 3 and length 2x + 1. a. CONCRETE Use algebra tiles to represent this situation. Use the model to find the width. b. SYMBOLIC Write an expression to represent the model. c. NUMERICAL Solve this problem algebraically using synthetic or long division. Does your concrete model check with your algebraic model? a. The width is x + 3. b. c. Yes, the concrete model checks with the algebraic model. 44. CHALLENGE If a polynomial is divided by a binomial and the remainder is 0, what does this tell you about the relationship between the binomial and the polynomial? If a polynomial divided by a binomial has no remainder, then the polynomial has two factors: the binomial and the quotient. 46. OPEN ENDED Write a quotient of two polynomials for which the remainder is 3. Sample answer: Begin by multiplying two binomials such as (x + 2)(x + 3) which simplifies to x 2 + 5x + 6. In order to get a remainder of 3 when divided, add 3 to the trinomial to get x 2 + 5x + 9. When divided, there will be a remainder of 3. The quotient of two polynomials is. esolutions Manual - Powered by Cognero Page 8
9 48. WRITING IN MATH Use the information at the beginning of the lesson to write assembly instruction using the division of polynomials to make a paper cover for your textbook. Sample answer: By dividing the area of the paper 140x x by the height of the book jacket 10x, the quotient of 14x + 6 provides the length of the book jacket. The front and back cover are each 6x units long and the spine is 2x units long. Then, subtracting 14x, we are left with 6 inches. Half of this length is the width of each flap. 50. SAT/ACT Which polynomial has degree 3? A x 3 + x 2 2x 4 B 2x 2 3x + 4 C 3x 3 D x 2 + x E 1 + x + x 3 To determine the degree of a polynomial, look at the term with the greatest exponent. The correct choice is E. 52. ( 4x 2 + 2x + 3) 3(2x 2 5x + 1) = F 2x 2 H 10x x G 10x 2 J 2x x The correct choice is H. Simplify. 54. (2y 3 3y + 8) + (3y 2 6y) 56. (c + d)(c d)(2c 3d) Use the FOIL method and then combine like terms. esolutions Manual - Powered by Cognero Page 9
10 58. (3ab 2 ) 2 (2a 2 b) 2 Solve each equation by completing the square. 60. x 2 + 6x + 2 = 0 esolutions Manual - Powered by Cognero Page 10
11 62. 2x 2 + 6x + 5 = 0 State the consecutive integers between which the zeros of each quadratic function are located. 64. The sign of f (x) changes between the x values between 0 and 1, and between 2 and 3. So the zeros of the quadratic function lies between 0 and 1, and between 2 and 3. esolutions Manual - Powered by Cognero Page 11
12 66. BUSINESS A landscaper can mow a lawn in 30 minutes and perform a small landscape job in 90 minutes. He works at most 10 hours per day, 5 days per week. He earns $35 per lawn and $125 per landscape job. He cannot do more than 3 landscape jobs per day. Find the combination of lawns mowed and completed landscape jobs per week that will maximize income. Then find the maximum income. Let M be the lawns mowed and L be the small landscaping jobs. Since the landscaper will do at most 3 landscaping jobs per day, L 15. Since he works 10 hours per day, 5 days a week, he can do at most 20 mowing jobs per day or 100 per week. So, M 100. If the landscaper does 3 landscaping jobs per day, he can only do 11 mowing jobs per day and work a total of 10 hours per day. So, for the week, if he does 15 landscaping jobs, he can mow 55 lawns. To maximize the earnings, write a function that relates the number and charge for each type of job. Since he charges $35 for each lawn mowed and $125 for each small landscaping job, F(M, L) = 35M + 125L. Next, substitute in the key values of M and L to determine the earnings for each combination of jobs. (M, L) F(M, L) = 35M + 125L F(M, L) (100, 0) F(M, L) = 35(100) + 125(0) 3500 (55, 15) F(M, L) = 35(55) + 125(15) 3800 (0, 15) F(M, L) = 35(0) + 125(15) landscape jobs and 55 lawns; $3800 Find each value if f (x) = 4x + 3, g(x) = x 2, and h(x) = 2x 2 2x g( 8) Substitute 8 for x in g(x). 70. f (c) Substitute c for x in f (x). esolutions Manual - Powered by Cognero Page 12
13 72. h(2b + 1) Substitute 2b + 1 for x in h(x). esolutions Manual - Powered by Cognero Page 13
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