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 11-5) 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 2-16 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 3-6q) 3q 1B. ( 4 t 5-5 t 2-12) 2 t 2 1C. (4 r r 4-2 r 2 ) 2r 1D. ( 6 w 3-3w) 4 w Lesson 11-5

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 2-2b - 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.mcgraw-hill.com 701

3 Real-World 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 2-4) (r - 1) 5B. ( x 4 + 2x 3 + 6x - 10) (x + 2) Check Your Understanding = Step-by-Step 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 2-3) (2h + 3) 11. (9 n 3-13n + 8) (3n - 1) 702 Lesson 11-5 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 2-18a) a 14. (5 q 3 + q) q 15. (6 n 2-12n + 3) 3n 16. (8 k 2-6) 2k 17. (9 m 2 + 5m) 6m 18. ( a 2 + a - 12) (a - 3) 19. ( x 2-6x - 16) (x + 2) 20. ( r 2-12r + 11) (r - 1) 21 ( k 2-5k - 24) (k - 8) 22. ( y 2-36) ( y 2 + 6y) 23. ( a 3-4 a 2 ) (a - 4) 24. ( c 3-27) (c - 3) 25. ( 4 t 2-1) (2t + 1) 26. (6 x x 2-60x + 39) (2x + 10) 27. (2 h h 2-3h - 12) (h + 4) Example GEOMETRY The area of a rectangle is ( x 3-4 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 2-3y + 1) (y - 2) 33. (4 n 2-3n + 6) (n - 2) 34. ( p 3-4 p 2 + 9) (p - 1) 35. ( t 3-2t - 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 2-3x - 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 4-6 y 3-9 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.mcgraw-hill.com 703

6 Virginia SOL Practice 51. Simplify 21 x 3-35 x 2. 7x A 3 x 2-5x C 3x - 5 B 4 x 2-6x 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 11-4) 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 11-3) x f(x) = 60. f(x) = x 2-3x - 4 x 2-6x + 8 x 2 - x f(x) = x 2 + 6x + 9 x SHADOWS A 25-foot 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 10-7) Solve each equation. Check your solution. (Lesson 10-4) 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 9-5) 67. v v + 20 = t 2-7t - 20 = y 2 - y - 4 = x = 28x n 2-7n - 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 9-2) a. Graph the quadratic function and determine its x-intercepts. b. What is the height of the arch? Skills Review Find each sum. (Lesson 7-5) 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 8-1) 76. 2, 4, , 6, , 12, , 18, 24 connected.mcgraw-hill.com 705

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