Why? _ v a There are different ways to simplify the expression. one fraction. term by 2a. = _ b 2


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1 Dividing Polynomials Then You divided rational expressions. (Lesson 115) Now 1Divide a polynomial by a monomial. 2Divide a polynomial by a binomial. Why? The equation below describes the distance d a horse travels when its initial velocity is 4 m/s, its final velocity is v m/s, and its acceleration is a m/ s 2. d = v a There are different ways to simplify the expression. Keep as Divide each one fraction. term by 2a. v = v 216 v = v 2 2a 2a 2a 2a a = v 2 2a  8 a Virginia i SOL A.2.b The student will perform operations on polynomials, including adding, subtracting, multiplying, and dividing polynomials. Divide Polynomials by Monomials To divide a polynomial by a monomial, 1 divide each term of the polynomial by the monomial. Example 1 Divide Polynomials by Monomials Find each quotient. a. (2 x x) 2x (2 x x) 2x = 2 x x 2x = 2 x 2 2x + 16x 2x x 8 Write as a fraction. Divide each term by 2x. = 2 x x 2x 2x 1 1 Divide out common factors. = x + 8 Simplify. b. ( b b  14) 3b ( b b  14) 3b = b b b = b 2 3b + 12b 3b b b = b b b 3b 3b 3 1 = b b 4 Write as a fraction. Divide each term by 3b. Divide out common factors. Simplify. 1A. (3 q 36q) 3q 1B. ( 4 t 55 t 212) 2 t 2 1C. (4 r r 42 r 2 ) 2r 1D. ( 6 w 33w) 4 w Lesson 115
2 Divide Polynomials by Binomials You can also divide polynomials by 2 binomials. When a polynomial can be factored and common factors can be divided out, write the division as a rational expression and simplify. Example 2 Divide a Polynomial by a Binomial Find ( h 2 + 9h + 18) (h + 6). ( h 2 + 9h + 18) (h + 6) = h 2 + 9h + 18 h + 6 Find each quotient. = (h + 3)(h + 6) h Write as a rational expression. Factor the numerator. (h + 3) (h + 6) = Divide out common factors. h = h + 3 Simplify. 2A. ( b 22b  15) (b + 3) 2B. ( x x + 24) (x + 8) If the polynomial cannot be factored or if there are no common factors by which to divide, you must use long division. Watch Out! Polynomials When using long division, be sure the dividend is written in standard form. That is, the terms are written so that the exponents decrease from left to right. y 2 + 4y y + y yes no Example 3 Use Long Division Find ( y 2 + 4y + 12) (y + 3) by using long division. Step 1 Divide the first term of the dividend, y 2, by the first term of the divisor, y. y y 2 y = y y + 3 y 2 + 4y + 12 () y 2 + 3y Multiply y and y + 3 1y + 12 Subtract. Bring down the 12. Step 2 Divide the first term of the partial dividend, 1y, by the first term of the divisor, y. y + 1 y + 3 y 2 + 4y + 12 () y 2 + 3y 1y + 12 Subtract. Bring down the 12. () y + 3 Multiply 1 and y Subtract. So, ( y 2 + 4y + 12) (y + 3) is y + 1 with a remainder of 9. This answer can be written as y y A. (3 x 2 + 9x  15) (x + 5) 3B. ( n 2 + 6n + 2) (n  2) connected.mcgrawhill.com 701
3 RealWorld Example 4 Divide Polynomials to Solve a Problem PARTIES The expression 5x represents the cost of renting a picnic shelter and food for x people. The total cost is divided evenly among all the people except for the two who bought decorations. Find (5x + 250) (x  2) to determine how much each person pays. 5 x  2 5x () 5x So, represents the amount each person pays. x GEOMETRY The area of a rectangle is (2 x x  1) square units, and the width is (x + 1) units. What is the length? When a dividend is written in standard form and a power is missing, add a term of that power with a coefficient of zero. Example 5 Insert Missing Terms Find ( c 3 + 5c  6) (c  1). c 2 + c + 6 c  1 c c 2 + 5c  6 Insert a c 2 term that has a coefficient of 0. () c 3  c 2 Multiply c 2 and c  1. c 2 + 5c Subtract. Bring down the 5c. () c 2  c Multiply c and c c  6 Subtract. Bring down the 6. () 6c  6 Multiply 6 and c Subtract. So, ( c 3 + 5c  6) (c  1) = c 2 + c A. (2 r r 24) (r  1) 5B. ( x 4 + 2x 3 + 6x  10) (x + 2) Check Your Understanding = StepbyStep Solutions begin on page R12. Examples 1 2Find each quotient. Example 4 1 (8 a a) 4a 2. (4 z 3 + 1) 2z 3. (12 n 3 6 n ) 6n 4. ( t 2 + 5t + 4) (t + 4) 5. ( x 2 + 3x  28) (x + 7) 6. ( x 2 + x  20) (x  4) x 7. CHEMISTRY The formula y = describes a mixture when x liters of a 50 + x 25% solution are added to a 90% solution. Find ( x) (50 + x). Examples 3 5Find each quotient. Use long division. 8. ( n 2 + 3n + 10) (n  1) 9. (4 y 2 + 8y + 3) (y + 2) 10. (4 h h 23) (2h + 3) 11. (9 n 313n + 8) (3n  1) 702 Lesson 115 Dividing Polynomials
4 Practice and Problem Solving Extra Practice begins on page 815. Examples 1 2Find each quotient. 12. (14 x 2 + 7x) 7x 13. ( a a 218a) a 14. (5 q 3 + q) q 15. (6 n 212n + 3) 3n 16. (8 k 26) 2k 17. (9 m 2 + 5m) 6m 18. ( a 2 + a  12) (a  3) 19. ( x 26x  16) (x + 2) 20. ( r 212r + 11) (r  1) 21 ( k 25k  24) (k  8) 22. ( y 236) ( y 2 + 6y) 23. ( a 34 a 2 ) (a  4) 24. ( c 327) (c  3) 25. ( 4 t 21) (2t + 1) 26. (6 x x 260x + 39) (2x + 10) 27. (2 h h 23h  12) (h + 4) Example GEOMETRY The area of a rectangle is ( x 34 x 2 ) square units, and the width is (x  4) units. What is the length? 29. MANUFACTURING The expression  n n represents the number of baseball caps produced by n workers. Find ( n n + 850) n to write an expression for average number of caps produced per person. Examples 3 5Find each quotient. Use long division. 30. ( b 2 + 3b  9) (b + 5) 31. ( a 2 + 4a + 3) (a  1) 32. (2 y 23y + 1) (y  2) 33. (4 n 23n + 6) (n  2) 34. ( p 34 p 2 + 9) (p  1) 35. ( t 32t  4) (t + 4) 36. (6 x x 2 + 9) (2x + 3) 37. (8 c 3 + 6c  5) (4c  2) B 38. GEOMETRY The volume of a prism with a triangular base is 10 w w 2 + 5w  2. The height of the prism is 2w + 1, and the height of the triangle is 5w  1. What is the measure of the base of the triangle? (Hint: V = Bh) 5w  1 2w + 1 Use long division to find the expression that represents the missing length. 39. A = x 23x  18? 40. A = 4x x x + 4 x  6? 41. Determine the quotient when x x + 14 is divided by x What is 14 y y 46 y 39 y y + 48 divided by 2y + 3? 43. FUNCTIONS Consider f(x) = 3x + 4 x  1. a. Rewrite the function as a quotient plus a remainder. Then graph the quotient, ignoring the remainder. b. Graph the original function using a graphing calculator. c. How are the graphs of the function and quotient related? d. What happens to the graph near the excluded value of x? connected.mcgrawhill.com 703
5 44. ROAD TRIP The first Ski Club van has been on the road for 20 minutes, and the second van has been on the road for 35 minutes. a. Write an expression for the amount of time that each van has spent on the road after an additional t minutes. b. Write a ratio for the first van s time on the road to the second van s time on the road and use long division to rewrite this ratio as an expression. Then find the ratio of the first van s time on the road to the second van s time on the road after 60 minutes, 200 minutes. 45 BOILING POINT The temperature at which water boils decreases by about 0.9 F for every 500 feet above sea level. The boiling point at sea level is 212 F. a. Write an equation for the temperature T at which water boils x feet above sea level. b. Mount Whitney, the tallest point in California, is 14,494 feet above sea level. At approximately what temperature does water boil on Mount Whitney? C 46. MULTIPLE REPRESENTATIONS In this problem, you will use picture models to help divide expressions. a. Analytical The first figure models Notice that the square is divided into seven equal parts. What are the quotient and the remainder? What division problem does the second figure model? b. Concrete Draw figures for and c. Verbal Do you observe a pattern in the previous exercises? Express this pattern algebraically. d. Analytical Use long division to find x 2 (x + 1). Does this result match your expression from part c? 6 6 H.O.T. Problems Use HigherOrder Thinking Skills 47. ERROR ANALYSIS Alvin and Andrea are dividing c 3 + 6c  4 by c + 2. Is either of them correct? Explain your reasoning. 704 Lesson 115 Dividing Polynomials Alvin c 2 + 4c 12 c + 2 c 3 + 6c 4 c 3 + 2c 2 4 c 2 4 4c 2 + 8c 12c 12c Andrea c 2 2c + 10 c + 2 c c 2 + 6c 4 c c 2 2 c 2 + 6c 2 c 2 4c 10c 4 10c CHALLENGE The quotient of two polynomials is 4 x 2 11x x x 2 + x + 2. What are the polynomials? 49. OPEN ENDED Write a division problem involving polynomials that you would solve by using long division. Explain your answer. 50. WRITING IN MATH Describe the steps to find ( w 22w  30) (w + 7).
6 Virginia SOL Practice 51. Simplify 21 x 335 x 2. 7x A 3 x 25x C 3x  5 B 4 x 26x D 5x EXTENDED RESPONSE The box shown is designed to hold rice. 9 cm 8 cm 5 cm a. How much rice would fit in the box? b. What is the area of the label on the box, if the label covers all surfaces? 53. Simplify x 2 + 7x + 12 x 2 + 5x + 6. F x + 4 H x + 2 G x + 4 x + 2 J x + 2 x Susana bought cards at 6 for $10. She decorated them and sold them at 4 for $10. She made $60 in profit. How many cards did she sell? A 25 C 60 B 53 D 72 A.2.b, A.4.f Spiral Review Find each product. (Lesson 114) x 3 8x 16 3ad x c 8 c 2 4 6d 57. t 2 (t  4)(t + 4) t  4 6t r  2 r Find the zeros of each function. (Lesson 113) x f(x) = 60. f(x) = x 23x  4 x 26x + 8 x 2  x f(x) = x 2 + 6x + 9 x SHADOWS A 25foot flagpole casts a shadow that is 10 feet long and a nearby building casts a shadow that is 26 feet long. How tall is the building? (Lesson 107) Solve each equation. Check your solution. (Lesson 104) 63. h = x + 3 = n = x  5 = 2 6 Solve each equation by using the Quadratic Formula. Round to the nearest tenth if necessary. (Lesson 95) 67. v v + 20 = t 27t  20 = y 2  y  4 = x = 28x n 27n  3 = w 2 = (7w + 3) 73. THEATER The drama club is building a backdrop using arches with a shape that can be represented by the function f(x) =  x 2 + 2x + 8, where x is the length of the arch in feet. The region under each arch is to be covered with fabric. (Lesson 92) a. Graph the quadratic function and determine its xintercepts. b. What is the height of the arch? Skills Review Find each sum. (Lesson 75) 74. (3 a 2 + 2a  12) + (8a a 2 ) 75. (2 c 3 + 3cd  d 2 ) + (5cd  2 c d 2 ) Find the least common multiple for each set of numbers. (Lesson 81) 76. 2, 4, , 6, , 12, , 18, 24 connected.mcgrawhill.com 705
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