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8-5 Dividing Polynomials

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8-5 Dividing Polynomials 8-5 Dividing Polynomials TEKS FOCUS VOCABULARY TEKS (7)(C) Determine the quotient of a • Synthetic division – Synthetic division is a process for dividing a polynomial of degree three and degree four polynomial by a linear expression x - a. You list the standard-form when divided by a polynomial of degree one coefficients (including zeros) of the polynomial, omitting all variables and of degree two. and exponents. You use a for the “divisor” and add instead of subtract throughout the process. TEKS (1)(A) Apply mathematics to problems arising in everyday life, society, and the Apply – use knowledge or information for a specific purpose, such as workplace. • solving a problem Additional TEKS (1)(G) ESSENTIAL UNDERSTANDING You can divide polynomials using steps that are similar to the long-division steps that you use to divide whole numbers. Key Concept The Division Algorithm for Polynomials You can divide polynomial P (x) by polynomial D (x) to get polynomial quotient Q (x) and polynomial remainder R (x). The result is P (x) = D (x)Q (x) + R (x). Q (x) D (x)∙P (x) ## # R (x) If R (x) = 0, then P (x) = D (x)Q (x) and D (x) and Q (x) are factors of P (x). To use long division, P (x) and D (x) should be in standard form with zero coefficients where appropriate. The process stops when the degree of the remainder, R (x), is less than the degree of the divisor, D (x). Theorem The Remainder Theorem If you divide a polynomial P (x) of degree n Ú 1 by x - a, then the remainder is P (a). PearsonTEXAS.com 363 Problem 1 Using Polynomial Long Division Use polynomial long division to divide 4x3 ∙ 3x2 ∙ 20x ∙ 46 by x2 ∙ 5. What is the quotient and remainder? 4x 3 x2 5∙4x 3 3x2 20x 46 Divide: 4x 4x. - + - - x2 = 4x 3 - 20x Multiply: 4x(x2 - 5) = 4x3 - 20x. 3x2 - 46 Subtract to get 3x2. Bring down -46. Repeat the process of dividing, multiplying, and subtracting. 4x + 3 2 x - 5∙4x3 + 3x2 - 20x - 46 4x3 - 20x 2 3x2 46 Divide: 3x 3 - x2 = 3x2 - 15 Multiply: 3(x2 - 5) = 3x2 - 15. -31 Subtract to get - 31. How can you check your result? The quotient is 4x + 3 with remainder -31. You can say: 4x ϩ 3, R Ϫ31. Show that (divisor)(quotient) + Check remainder = dividend. 2 3 2 (x - 5)(4x + 3) - 31 = (4x + 3x - 20x - 15) - 31 Multiply (x2 - 5)(4x + 3). = 4x3 + 3x2 - 20x - 46 ✔ Simplify. Problem 2 TEKS Process Standard (1)(G) Using Polynomial Long Division to Check Factors A Use polynomial long division to divide P(x) ∙ 3x4 ∙ 4x 3 ∙ 12x 2 ∙ 5 by x 2 ∙ 1. Is x 2 ∙ 1 a factor of P(x)? 3x 2 - 4x + 9 x 2 + 0x + 1∙3x 4 - 4x 3 + 12x 2 + 0x + 5 Include 0x terms. 3x 4 + 0x 3 + 3x 2 -4x 3 + 9x 2 + 0x -4x 3 + 0x 2 - 4x 9x 2 + 4x + 5 The degree of the remainder is 9x 2 + 0x + 9 less than the degree of the 4x - 4 divisor. Stop! The remainder is not zero. x2 + 1 does not divide 3x4 - 4x3 + 12x2 + 5 evenly and is not a factor of P(x). continued on next page ▶ 364 Lesson 8-5 Dividing Polynomials Problem 2 continued B Is x ∙ 2 a factor of P (x) ∙ x4 ∙ 16? If it is, write P (x) as a product of two factors. Can you use the Factor Theorem to Step 1 Use the Factor Theorem to determine if x - 2 is a factor of x4 - 16. help answer this 4 question? P (2) = 2 - 16 Yes; recall that if = 16 - 16 P(a) = 0, then x - a is a factor of P(x). = 0 Since P (2) = 0, x - 2 is a factor of P (x). Step 2 Use polynomial long division to find the other factor. x3 + 2x2 + 4x + 8 x - 2∙x4 + 0x3 + 0x2 + 0x - 16 x4 - 2x3 2x3 + 0x2 2x3 - 4x2 4x2 + 0x 4x2 - 8x 8x - 16 8x - 16 0 P (x) = (x - 2)(x 3 + 2x 2 + 4x + 8) Problem 3 Using Synthetic Division 4 3 2 To divide by x ∙ 2 Use synthetic division to divide x ∙ 14x ∙ 51x ∙ 54x ∙ 110 by x ∙ 2. what number do you What is the quotient and remainder? use for the synthetic divisor? Step 1 Reverse the sign of +2. Write Step 2 Bring down the first coefficient. x + 2 = x - (-2), so the coefficients of the polynomial. -2 1 -14 51 -54 -110 use - 2. -2 1 -1 4 5 1 -5 4 -1 1 0 ; 1 ; Step 3 Multiply the coefficient by the divisor. Step 4 Continue multiplying and adding Add to the next coefficient. through the last coefficient. -2 1 -14 51 -54 -110 -2 1 -14 51 -54 -110 -2 -2 32 -166 440 ; ; 1 -16 1 -16 83 -220 330 The quotient is x3 - 16x2 + 83x - 220, R 330. PearsonTEXAS.com 365 Problem 4 TEKS Process Standard (1)(A) Using Synthetic Division to Solve a Problem Crafts The polynomial x3 7x2 38x 240 expresses the volume, in cubic inches, of the shadow box shown. A What are the dimensions of the box? (Hint: The length is greater than the height (or depth).) How can you use the picture to help solve -5 1 7 -38 -240 the problem? -5 -10 240 The picture gives the width ; of the box. Remember 1 2 -48 0 for a rectangular prism, x2 + 2x - 48 = (x - 6)(x + 8) V = / * w * h. So, x3 + 7x2 - 38x - 240 = (x + 5)(x2 + 2x - 48) = (x + 5)(x - 6)(x + 8) x + 5 The length, width, and height (or depth) of the box are (x + 8) in., (x + 5) in., and (x - 6) in., respectively. B If the width of the box is 15 in., what are the other two dimensions? The width of the box is x + 5. So if x + 5 = 15, then x = 10. Substitute for x to find the length and height (or depth). Length: x + 8 = 10 + 8 = 18 in. Height: x - 6 = 10 - 6 = 4 in. Problem 5 Evaluating a Polynomial Given that P (x) x4 2x2 x 122, what is P (3)? Is there a way to find P(3) without By the Remainder Theorem, P (3) is the remainder when you 1 8 2 substituting? ....... divide P (x) by x - 3. 0 0 0 0 0 0 0 Use synthetic division. 1 1 1 1 1 1 1 P(3) is the remainder. 3 1 0 2 1 122 2 2 2 2 2 2 2 - - 3 3 3 3 3 3 3 3 -9 21 60 4 4 4 4 4 4 4 ; 5 5 5 5 5 5 5 1 3 7 20 182 6 6 6 6 6 6 6 7 7 7 7 7 7 7 8 8 8 8 8 8 8 P (3) = 182. 9 9 9 9 9 9 9 366 Lesson 8-5 Dividing Polynomials NLINE O H K Scan page for a Virtual Nerd™ tutorial video. O PRACTICE and APPLICATION EXERCISES M R E W O Divide using long division. Check your answers. 1. 3x2 + 7x - 20 , (x + 4) 2. x3 + 3x2 - x + 2 , (x - 1) For additional support when 3. 2x3 - 3x2 - 18x - 8 , (x - 4) 4. x3 + 5x2 - 4x - 20 , (x2 - 4) completing your homework, 1 2 1 2 go to PearsonTEXAS.com. Divide. 1 2 1 2 5. 2x3 + 9x2 + 14x + 5 , (2x2 + 1) 6. x4 + 3x2 + x + 4 , (x + 3) 7. x4 + 4x3 - x - 4 , (x2 - 1) 8. 3x4 - 5x3 + 2x2 + 3x - 2 , (3x - 2) 1 2 1 2 Determine whether each binomial is a factor of x3 4x2 x 6. 1 2 1 2 9. x + 1 10. x + 2 11. x + 3 12. x - 3 Divide using synthetic division. 13. x3 + 3x2 - x - 3 , (x - 1) 14. x3 - 4x2 + 6x - 4 , (x - 2) 15. x3 - 7x2 - 7x + 20 , (x + 4) 16. x3 - 3x2 - 5x - 25 , (x - 5) 1 2 1 2 17. x2 + 3 , (x - 1) 18. 3x3 + 17x2 + 21x - 9 , (x + 3) 1 2 1 2 19. x3 + 27 , (x + 3) 20. 6x2 - 8x - 2 , (x - 1) 1 2 1 2 Use synthetic division and the given factor to completely factor each polynomial1 2 function. 1 2 21. y = x3 + 2x2 - 5x - 6; (x + 1) 22. y = x3 - 4x2 - 9x + 36; (x + 3) 23. Apply Mathematics (1)(A) The volume, in cubic inches, of the decorative box shown can be expressed as the product of the lengths of its sides as V (x) = x3 + x2 - 6x. What linear expressions with integer coefficients represent the length and height of the box? x Use synthetic division and the Remainder Theorem to find P (a). 24. P (x) = x3 + 4x2 - 8x - 6; a = -2 25. P (x) = x3 + 4x2 + 4x; a = -2 26. P (x) = x3 - 7x2 + 15x - 9; a = 3 27. P (x) = x3 + 7x2 + 4x; a = -2 3 2 3 2 1 28. P (x) = 6x - x + 4x + 3; a = 3 29. P (x) = 2x - x + 10x + 5; a = 2 30. P (x) = 2x3 + 4x2 - 10x - 9; a = 3 31. P (x) = 2x4 + 6x3 + 5x2 - 45; a = -3 PearsonTEXAS.com 367 32. Select Techniques to Solve Problems (1)(C) Your friend multiplies x + 4 by a quadratic polynomial and gets the result x3 - 3x2 - 24x + 30. The teacher says that everything is correct except for the constant term. Find the quadratic polynomial that your friend used. What is the correct result of multiplication? 33.
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