Add, Subtract, and Multiply Polynomials

Transcription

1 TEKS 5.3 a.2, 2A.2.A; P.3.A, P.3.B Add, Subtract, and Multiply Polynomials Before You evaluated and graphed polynomial functions. Now You will add, subtract, and multiply polynomials. Why? So you can model collegiate sports participation, as in Ex. 63. Key Vocabulary like terms, p. 12 To add or subtract polynomials, add or subtract the coefficients of like terms. You can use a vertical or horizontal format. E XAMPLE 1 Add polynomials vertically and horizontally a. Add 2x 3 2 5x 2 1 3x 2 9 and x 3 1 6x in a vertical format. b. Add 3y 3 2 2y 2 2 7y and 24y 2 1 2y 2 5 in a horizontal format. REVIEW SIMPLIFYING For help with simplifying expressions, see p. 10. a. 2x 3 2 5x 2 1 3x x 3 1 6x x 3 1 x 2 1 3x 1 2 b. (3y 3 2 2y 2 2 7y) 1 (24y 2 1 2y 2 5) 5 3y 3 2 2y 2 2 4y 2 2 7y 1 2y y 3 2 6y 2 2 5y 2 5 E XAMPLE 2 Subtract polynomials vertically and horizontally a. Subtract 3x 3 1 2x 2 2 x 1 7 from 8x 3 2 x 2 2 5x 1 1 in a vertical format. b. Subtract 5z 2 2 z 1 3 from 4z 2 1 9z 2 12 in a horizontal format. a. Align like terms, then add the opposite of the subtracted polynomial. 8x 3 2 x 2 2 5x 1 1 8x 3 2 x 2 2 5x (3x 3 1 2x 2 2 x 1 7) 1 23x 3 2 2x 2 1 x 2 7 5x 3 2 3x 2 2 4x 2 6 b. Write the opposite of the subtracted polynomial, then add like terms. (4z 2 1 9z 2 12) 2 (5z 2 2 z 1 3) 5 4z 2 1 9z z 2 1 z z 2 2 5z 2 1 9z 1 z z z 2 15 GUIDED PRACTICE for Examples 1 and 2 Find the sum or difference. 1. (t 2 2 6t 1 2) 1 (5t 2 2 t 2 8) 2. (8d d 3 ) 2 (d d 2 2 4) 346 Chapter 5 Polynomials and Polynomial Functions

2 MULTIPLYING POLYNOMIALS To multiply two polynomials, you multiply each term of the first polynomial by each term of the second polynomial. E XAMPLE 3 Multiply polynomials vertically and horizontally a. Multiply 22y 2 1 3y 2 6 and y 2 2 in a vertical format. b. Multiply x 1 3 and 3x 2 2 2x 1 4 in a horizontal format. a. 22y 2 1 3y y 2 2 4y 2 2 6y 1 12 Multiply 22y 2 1 3y 2 6 by y 3 1 3y 2 2 6y Multiply 22y 2 1 3y 2 6 by y. 22y 3 1 7y y 1 12 Combine like terms. b. (x 1 3)(3x 2 2 2x 1 4) 5 (x 1 3)3x 2 2 (x 1 3)2x 1 (x 1 3)4 5 3x 3 1 9x 2 2 2x 2 2 6x 1 4x x 3 1 7x 2 2 2x 1 12 E XAMPLE 4 Multiply three binomials Multiply x 2 5, x 1 1, and x 1 3 in a horizontal format. (x 2 5)(x 1 1)(x 1 3) 5 (x 2 2 4x 2 5)(x 1 3) 5 (x 2 2 4x 2 5)x 1 (x 2 2 4x 2 5)3 5 x 3 2 4x 2 2 5x 1 3x x x 3 2 x x 2 15 PRODUCT PATTERNS Some binomial products occur so frequently that it is worth memorizing their patterns. You can verify these product patterns by multiplying. KEY CONCEPT For Your Notebook Special Product Patterns Sum and Difference Example (a 1 b)(a 2 b) 5 a 2 2 b 2 (x 1 4)(x 2 4) 5 x AVOID ERRORS In general, (a 6 b) 2 Þ a 2 6 b 2 and (a 6 b) 3 Þ a 3 6 b 3. Square of a Binomial Example (a 1 b) 2 5 a 2 1 2ab 1 b 2 (a 2 b) 2 5 a 2 2 2ab 1 b 2 (y 1 3) 5 y 2 1 6y 1 9 (3z 2 2 5) 2 5 9z z Cube of a Binomial Example (a 1 b) 3 5 a 3 1 3a 2 b 1 3ab 2 1 b 3 (a 2 b) 3 5 a 3 2 3a 2 b 1 3ab 2 2 b 3 (x 1 2) 5 x 3 1 6x x 1 8 (p 2 3) 3 5 p 3 2 9p p Add, Subtract, and Multiply Polynomials 347

3 E XAMPLE 5 Use special product patterns a. (3t 1 4)(3t 2 4) 5 (3t) Sum and difference 5 9t b. (8x 2 3) 2 5 (8x) 2 2 2(8x)(3) Square of a binomial 5 64x x 1 9 c. (pq 1 5) 3 5 (pq) 3 1 3(pq) 2 (5) 1 3(pq)(5) Cube of a binomial 5 p 3 q p 2 q pq GUIDED PRACTICE for Examples 3, 4, and 5 Find the product. 3. (x 1 2)(3x 2 2 x 2 5) 4. (a 2 5)(a 1 2)(a 1 6) 5. (xy 2 4) 3 E XAMPLE 6 Use polynomial models PETROLEUM Since 1980, the number W (in thousands) of United States wells producing crude oil and the average daily oil output per well O (in barrels) can be modeled by W t t and O t where t is the number of years since Write a model for the average total amount T of crude oil produced per day. What was the average total amount of crude oil produced per day in 2000? DETERMINE SIGNIFICANT DIGITS When multiplying models, round your result so that its terms have the same number of significant digits as the model with the fewest number of significant digits. To find a model for T, multiply the two given models t t t t t t t t t t t Oil refinery in Long Beach, California c Total daily oil output can be modeled by T t t t where T is measured in thousands of barrels. By substituting t 5 20 into the model, you can estimate that the average total amount of crude oil produced per day in 2000 was about 5570 thousand barrels, or 5,570,000 barrels. GUIDED PRACTICE for Example 6 6. INDUSTRY The models below give the average depth D (in feet) of new wells drilled and the average cost per foot C (in dollars) of drilling a new well. In both models, t represents the number of years since Write a model for the average total cost T of drilling a new well. D5 109t and C t t Chapter 5 Polynomials and Polynomial Functions

4 5.3 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS1 for Exs. 11, 21, and 61 5 TAKS PRACTICE AND REASONING Exs. 15, 47, 56, 63, 65, and VOCABULARY When you add or subtract polynomials, you add or subtract the coefficients of?. 2. WRITING Explain how a polynomial subtraction problem is equivalent to a polynomial addition problem. EXAMPLES 1 and 2 on p. 346 for Exs ADDING AND SUBTRACTING POLYNOMIALS Find the sum or difference. 3. (3x 2 2 5) 1 (7x 2 2 3) 4. (x 2 2 3x 1 5) 2 (24x 2 1 8x 1 9) 5. (4y 2 1 9y 2 5) 2 (4y 2 2 5y 1 3) 6. (z 2 1 5z 2 7) 1 (5z z 2 6) 7. (3s 3 1 s) 1 (4s 3 2 2s 2 1 7s 1 10) 8. (2a 2 2 8) 2 (a 3 1 4a a 1 4) 9. (5c 2 1 7c 1 1) 1 (2c 3 2 6c 1 8) 10. (4t t 2 1 4t) 2 (27t 2 2 5t 1 8) 11. (5b 2 6b 3 1 2b 4 ) 2 (9b 3 1 4b 4 2 7) 12. (3y 2 2 6y y) 1 (5y 4 2 6y 3 1 4y) 13. (x 4 2 x 3 1 x 2 2 x 1 1) 1 (x 1 x x 2 ) 14. (8v 4 2 2v 2 1 v 2 4) 2 (3v v 2 1 8v) 15. TAKS REASONING What is the result when 2x 4 2 8x 2 2 x 1 10 is subtracted from 8x 4 2 4x 3 2 x 1 2? A 26x 4 1 4x 3 2 8x B 6x 4 2 4x 3 1 8x C 10x 4 2 8x 3 2 4x D 6x 4 1 4x 3 2 2x 2 8 EXAMPLE 3 on p. 347 for Exs MULTIPLYING POLYNOMIALS Find the product of the polynomials. 16. x(2x 2 2 5x 1 7) 17. 5x 2 (6x 1 2) 18. (y 2 7)(y 1 6) 19. (3z 1 1)(z 2 3) 20. (w 1 4)(w 2 1 6w 2 11) 21. (2a 2 3)(a a 2 2) 22. (5c 2 2 4)(2c 2 1 c 2 3) 23. (2x 2 1 4x 1 1)(x 2 2 8x 1 3) 24. (2d 2 1 4d 1 3)(3d 2 2 7d 1 6) 25. (3y 2 1 6y 2 1)(4y y 2 5) ERROR ANALYSIS Describe and correct the error in simplifying the expression. 26. (x 2 2 3x 1 4) 2 (x 3 1 7x 2 2) 27. (2x 2 7) 3 5 (2x) x 2 2 3x x 3 1 7x x x 3 1 x 2 1 4x 1 2 EXAMPLE 4 on p. 347 for Exs MULTIPLYING THREE BINOMIALS Find the product of the binomials. 28. (x 1 4)(x 2 6)(x 2 5) 29. (x 1 1)(x 2 7)(x 1 3) 30. (z 2 4)(2z 1 2)(z 1 8) 31. (a 2 6)(2a 1 5)(a 1 1) 32. (3p 1 1)(p 1 3)(p 1 1) 33. (b 2 2)(2b 2 1)(2b 1 1) 34. (2s 1 1)(3s 2 2)(4s 2 3) 35. (w 2 6)(4w 2 1)(23w 1 5) 36. (4x 2 1)(22x 2 7)(25x 2 4) 37. (3q 2 8)(29q 1 2)(q 2 2) 5.3 Add, Subtract, and Multiply Polynomials 349

5 EXAMPLE 5 on p. 348 for Exs SPECIAL PRODUCTS Find the product. 38. (x 1 5)(x 2 5) 39. (w 2 9) (y 1 4) (2c 1 5) (3t 2 4) (5p 2 3)(5p 1 3) 44. (7x 2 y) (2a 1 9b)(2a 2 9b) 46. (3z 1 7y) TAKS REASONING Which expression is equivalent to (3x 2 2y) 2? A 9x 2 2 4y 2 B 9x 2 1 4y 2 C 9x xy 1 4y 2 D 9x xy 1 4y 2 GEOMETRY Write the figure s volume as a polynomial in standard form. 48. V 5 lwh 3x 1 1 x x V 5 πr 2 h 2x 1 3 x V 5 s 3 x V 5 1 } 3 Bh 3x 1 4 2x 2 3 2x 2 3 SPECIAL PRODUCTS Verify the special product pattern by multiplying. 52. (a 1 b)(a 2 b) 5 a 2 2 b (a 1 b) 2 5 a 2 1 2ab 1 b (a 1 b) 3 5 a 3 1 3a 2 b 1 3ab 2 1 b (a 2 b) 3 5 a 3 2 3a 2 b 1 3ab 2 2 b TAKS REASONING Let p(x) 5 x 4 2 7x 1 14 and q(x) 5 x a. What is the degree of the polynomial p(x) 1 q(x)? b. What is the degree of the polynomial p(x) 2 q(x)? c. What is the degree of the polynomial p(x) p q(x)? d. In general, if p(x) and q(x) are polynomials such that p(x) has degree m, q(x) has degree n, and m > n, what are the degrees of p(x) 1 q(x), p(x) 2 q(x), and p(x) p q(x)? 57. FINDING A PATTERN Look at the following polynomial factorizations. x (x 2 1)(x 1 1) x (x 2 1)(x 2 1 x 1 1) x (x 2 1)(x 3 1 x 2 1 x 1 1) a. Factor x and x into the product of x 2 1 and another polynomial. Check your answers by multiplying. b. In general, how can x n 2 1 be factored? Show that this factorization works by multiplying the factors. 58. CHALLENGE Suppose f(x) 5 (x 1 a)(x 1 b)(x 1 c)(x 1 d). If f(x) is written in standard form, show that the coefficient of x 3 is the sum of a, b, c, and d, and the constant term is the product of a, b, c, and d WORKED-OUT SOLUTIONS on p. WS1 5 TAKS PRACTICE AND REASONING

6 PROBLEM SOLVING EXAMPLE 6 on p. 348 for Exs HIGHER EDUCATION Since 1970, the number (in thousands) of males M and females F attending institutes of higher education can be modeled by M t t t and F t t t where t is the number of years since Write a model for the total number of people attending institutes of higher education. 60. ELECTRONICS From 1999 to 2004, the number of DVD players D (in millions) sold in the United States and the average price per DVD player P (in dollars) can be modeled by D t and P t t where t is the number of years since Write a model for the total revenue R from DVD sales. According to the model, what was the total revenue in 2002? 61. BICYCLING The equation P sF gives the power P (in horsepower) needed to keep a certain bicycle moving at speed s (in miles per hour), where F is the force (in pounds) of road and air resistance. On level ground, the equation F s models the force F. Write a model (in terms of s only) for the power needed to keep the bicycle moving at speed s on level ground. How much power is needed to keep the bicycle moving at 10 miles per hour? at classzone.com 62. MULTI-STEP PROBLEM A dessert is made by taking a hemispherical mound of marshmallow on a 0.5 centimeter thick cookie and covering it with a chocolate shell 1 centimeter thick. Use the diagrams to write two polynomial functions in standard form: M(r) for the combined volume of the marshmallow plus cookie, and D(r) for the volume of the entire dessert. Then use M(r) and D(r) to write a function C(r) for the volume of the chocolate. 63. TAKS REASONING From 1997 to 2002, the number of NCAA lacrosse teams for men L m and women L w, as well as the average size of a men s team S m and a women s team S w, can be modeled by L m t and S m t t t L w t and S w t t t where t is the number of years since Write a model for the total number of people N on NCAA lacrosse teams. Explain how you obtained your model. 5.3 Add, Subtract, and Multiply Polynomials 351

7 64. CHALLENGE From 1970 to 2002, the circulation C (in millions) of Sunday newspapers in the United States can be modeled by C t t t where t is the number of years since Rewrite C as a function of s, where s is the number of years since MIXED REVIEW FOR TAKS TAKS PRACTICE at classzone.com REVIEW Lesson 2.5; TAKS Workbook REVIEW Skills Review Handbook p. 1006; TAKS Workbook 65. TAKS PRACTICE The table shows the total cost y of heating oil. Which equation best represents the total cost of the heating oil as a function of the number of gallons x? TAKS Obj. 1 A x y B y x C x 5 1.5y D y 5 1.5x 66. TAKS PRACTICE A student is making a circle graph of the results of a survey that asked what people s favorite sport is. What central angle should be used for the section representing basketball? TAKS Obj. 9 F 358 G 1058 H 1268 J 2348 Number of gallons (x) Activity Number of people Basketball 350 Soccer 210 Softball or Baseball 200 Other 240 Total cost (y) 50 $ $ $750 QUIZ for Lessons Evaluate the expression. (p. 330) p (2 4 ) } } Simplify the expression. (p. 330) 5. (x 4 y 22 )(x 23 y 8 ) 6. (a 2 b 25 ) x 3 y 7 } x 24 y 0 8. c 3 d 22 } c 5 d 21 Graph the polynomial function. (p. 337) 9. g(x) 5 2x 3 2 3x h(x) 5 x 4 2 4x f(x) 522x 3 1 x Perform the indicated operation. (p. 346) 12. (x 3 1 x 2 2 6) 2 (2x 2 1 4x 2 8) 13. (23x 2 1 4x 2 10) 1 (x 2 2 9x 1 15) 14. (x 2 5)(x 2 2 5x 1 7) 15. (x 1 3)(x 2 6)(3x 2 1) 16. NATIONAL DEBT On July 21, 2004, the national debt of the United States was about $7,282,000,000,000. The population of the United States at that time was about 294,000,000. Suppose the national debt was divided evenly among everyone in the United States. How much would each person owe? (p. 330) 352 EXTRA PRACTICE for Lesson 5.3, p ONLINE QUIZ at classzone.com