8-5 Dividing Polynomials

Home , Degree of a polynomial, Polynomial long division

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).

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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?

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. Display Mathematical Ideas (1)(G) A student used synthetic division to divide x3 - x2 - 2x by x + 1. Describe and correct the error shown. 1 1 -1 -2 1 0 34. Connect Mathematical Ideas (1)(F) When a polynomial is divided 1 0 -2 by (x - 5), the quotient is 5x2 + 3x + 12 with remainder 7. Find the polynomial. 1 3 2 35. Apply Mathematics (1)(A) The expression 3(x + 5x + 8x + 4) represents the volume of a square pyramid. The expression x + 1 represents the height of the pyramid. What expression represents the side length of the base? 1 (Hint: The formula for the volume of a pyramid is V = 3Bh.) 36. Analyze Mathematical Relationships (1)(F) Divide. Look for patterns in your answers. a. x2 - 1 , (x - 1) b. x3 - 1 , (x - 1) c. x4 1 (x 1) d. Using the patterns, factor x5 1. 1 - 2 , - 1 2 - 37. Select Tools to Solve Problems (1)(C) The remainder from the division of the 1 2 polynomial x3 + ax2 + 2ax + 5 by x + 1 is 3. Find a. 38. Use synthetic division to find (x2 + 4) , (x - 2i). 39. Display Mathematical Ideas (1)(G) Suppose 3, -1, and 5 are zeros of a cubic polynomial function f (x). What is the sign of f (1) # f (4)? (Hint: Sketch the graph; consider all possibilities.)

TEXAS Test Practice

40. What is the remainder when x2 - 5x + 7 is divided by x + 1? A. 1 B. 3 C. 11 D. 13 41. What is the least degree of a polynomial that has a zero of multiplicity 3 at 1, a zero of multiplicity 1 at 0, and a zero of multiplicity 2 at 2? F. 3 G. 4 H. 5 J. 6

42. The equation y = 0.17x relates your weight on the Moon y to your weight on Earth x in pounds. If Al weighs 130 lb on Earth, what would he weigh on the Moon? A. 22.1 lb B. 92.3 lb C. 130 lb D. 764.7 lb

43. The formula for the area of a circle is A = pr 2. Solve the equation for r. If the area of a circle is 78.5 cm2, what is the radius? Use 3.14 for p.

368 Lesson 8-5 Dividing Polynomials