Long Division of Polynomials

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1 SECTION 4.3 DIVI DI NG P OLYNOM IALS: LONG DIVISION AN D SYNTH ETIC DIVISION SKI LLS OBJ ECTIVES Divide polynomials with long division. Divide polynomials with synthetic division. CONCE PTUAL OBJ ECTIVES Extend long division of real numbers to polynomials. Understand when synthetic division can be used To divide polynomials, we rely on the technique we use for dividing real numbers. For example, if you were asked to divide 6542 by 2, the long division method used is illustrated in the margin. This solution can be written two ways: 3 R or In this example, the dividend is 6542, the divisor is 2, the quotient is 3, and the remainder is. We employ a similar technique (dividing the leading terms) when dividing polynomials. Long Division of Polynomials Let s start with an example whose answer we already know. We know that a quadratic expression can be factored into the product of two linear factors: x 2 4x 5 (x 5)(x ). Therefore, if we divide both sides of the equation by (x ), we get x 2 + 4x - 5 = x + 5 We can state this by saying x 2 4x 5 divided by x is equal to x 5. Confirm this statement by long division: Note that although this is standard division notation, the dividend and the divisor are both polynomials that consist of multiple terms. The leading terms of each algebraic expression will guide us. WORDS MATH Q: x times what quantity gives x 2? A: x x Multiply x(x ) x 2 x. Subtract (x 2 x) from x 2 4x 5. Note: (x 2 x) x 2 x. x x 2 x Bring down the -5. 5x - 5 x -x 2 + x 40

2 4.3 Dividing Polynomials: Long Division and Synthetic Division 4 WORDS Q: x times what quantity is 5x? A: 5 Multiply 5() 5x - 5. MATH -x 2 + x 5x - 5 5x 5 x + 5 x + 5 -x 2 + x 5x - 5 Subtract (5x 5). -5x + 5 Note: (5x 5) 5x 5. 0 As expected, the remainder is 0. By long division we have shown that Technology Tip A graphing utility can be used to check (x 2 5x 6)(2x ) 2x 3 9x 2 7x 6 using their graphs. x 2 + 4x - 5 = x + 5 Check: Multiplying the equation by x yields x 2 4x 5 (x 5)(x ), which we knew to be true. EXAM PLE Divide 2x 3 9x 2 7x 6 by 2x. Multiply: x 2 (2x ). Subtract: Bring down the 7x. Multiply: 5x(2x ). Subtract: Bring down the 6. Multiply: 6(2x ). Dividing Polynomials Using Long Division; Zero Remainder Subtract. 0 Quotient: Check: (2x )(x 2 5x 6) 2x 3 9x 2 7x 6. x 2-5x + 6 2x + 2x 3-9x 2 + 7x + 6 -(2x 3 x 2 ) -0x 2 + 7x -( 0x 2 5x) -(2x 6) x 2-5x + 6 2x + 6 Note: Since the divisor cannot be equal to zero, 2x Z 0, then we say xz. 2 Notice that the graphs are the same. YOU R TU R N Divide 4x 3 3x 2 2x 5 by 4x 5. Answer: x 2 2x 3, remainder 0. Why are we interested in dividing polynomials? Because it helps us find zeros of polynomials. In Example, using long division, we found that 2x 3 9x 2 7x 6 (2x )(x 2 5x 6)

3 42 CHAPTE R 4 Polynomial and Rational Functions Factoring the quadratic expression enables us to write the cubic polynomial as a product of three linear factors: 2x 3 9x 2 7x 6 (2x )(x 2 5x 6) (2x )(x 3)(x 2) Set the value of the polynomial equal to zero, (2x )(x 3)(x 2) 0, and solve for x. The zeros of the polynomial are 2, 2, and 3. In Example and in the Your Turn, the remainder was 0. Sometimes there is a nonzero remainder (Example 2). EXAM PLE 2 Divide 6x 2 x 2 by x. Multiply 6x(x ). Subtract and bring down 2. Multiply 7(x ). Subtract and identify the remainder. Dividing Polynomials Using Long Division; Nonzero Remainder 6x - 7 x + 6x 2 - x - 2 -(6x 2 + 6x) -7x - 2 -(-7x - 7) + 5 Answer: 2x 2 3x R: 4 or 2x Dividend Quotient Remainder 6x 2 x 2 x Divisor Check: Multiply the quotient and remainder by x. = 6x x Divisor (6x 7)(x ) 6x 2 x 7 5 The result is the dividend. 6x 2 x 2 x Z - 5 (x + ) # (x + ) YOU R TU R N Divide 2x 3 x 2 4x 3 by x. In general, when a polynomial is divided by another polynomial, we express the result in the following form: P(x) d(x) = Q(x) + r(x) d(x) where P(x) is the dividend, d(x) Z 0 is the divisor, Q(x) is the quotient, and r(x) is the remainder. Multiplying this equation by the divisor, d(x), leads us to the division algorithm. THE DIVISION ALGORITHM If P(x) and d(x) are polynomials with d(x) Z 0, and if the degree of P(x) is greater than or equal to the degree of d(x), then unique polynomials Q(x) and r(x) exist such that P(x) = d(x)# Q(x) + r(x) If the remainder r(x) 0, then we say that d(x) divides P(x) and that d(x) and Q(x) are factors of P(x).

4 4.3 Dividing Polynomials: Long Division and Synthetic Division 43 E X A M P L E 3 Divide x 3 8 by x 2. Insert 0x 2 0x for placeholders. Multiply x 2 (x 2) x 3 2x 2. Subtract and bring down 0x. Multiply 2x(x 2) 2x 2 4x. Subtract and bring down 8. Multiply 4(x 2) 4x 8. Subtract and get remainder 0. Long Division of Polynomials with Missing Terms Since the remainder is 0, x 2 is a factor of x 3 8. x 3-8 x - 2 = x2 + 2x + 4, x Z 2 x 2 + 2x + 4 x - 2 x 3 + 0x 2 0x - 8 -(x 3-2x 2 ) 2x 2 + 0x -(2x 2-4x) 4x (4x - 8) 0 Check: x 3 8 (x 2 2x 4)(x 2) x 3 2x 2 4x 2x 2 4x 8 x 3 8 YOU R TU R N Divide x 3 by x. Answer: x 2 x EXAM PLE 4 Divide 3x 4 2x 3 x 2 4 by x 2. Insert 0x as a placeholder in both the divisor and the dividend. Multiply 3x 2 (x 2 0x ). Subtract and bring down 0x. Multiply 2x(x 2 0x ). Subtract and bring down 4. Multiply 2(x 2 2x ). Subtract and get remainder 2x 6. Long Division of Polynomials 3x 2 + 2x - 2 x 2 + 0x + 3x 4 + 2x 3 + x 2 + 0x + 4 -A3x 4 + 0x 3 + 3x 2 B 2x 3-2x 2 + 0x -(2x 3 + 0x 2 + 2x) -2x 2-2x + 4 -(-2x 2 + 0x - 2) -2x + 6 3x 4 + 2x 3 + x x 2 + = 3x 2 + 2x x + 6 x 2 + Answer: 2x x2 + 8x + 36 x 3-3x - 4 YOU R TU R N Divide 2x 5 3x 2 2 by x 3 3x 4.

5 44 CHAPTE R 4 Polynomial and Rational Functions In Examples through 4 the dividends, divisors, and quotients were all polynomials with integer coefficients. In Example 5, however, the resulting quotient has rational (noninteger) coefficients. EXAM PLE 5 Divide 8x 4-5x 3 + 7x - 2 by 2x 2 +. Insert 0x 2 as a placeholder in the dividend and 0x as a placeholder in the divisor. Multiply 4x 2 (2x 2 + 0x + ). Subtract and bring down remaining terms. Multiply x(2x2 + 0x + ). Subtract and bring down remaining terms. Multiply -2(2x 2 + 0x + ). Subtract and bring down the remainder Long Division of Polynomials Resulting in Quotients with Rational Coefficients 8x 4-5x 3 + 7x - 2 2x x. = 4x x x x - 2 2x 2 + 0x + 8x 4-5x 3 + 0x 2 + 7x - 2 -(8x 4 + 0x 3 + 4x 2 ) - 5x 3-4x 2 + 7x -(-5x 3 + 0x x) 9 2 x 2x x x - 2 -(-4x 2 + 0x - 2) 9 2 x Answer: 5x x x x 2 - YOU R TU R N Divide 0x 4-3x 3 + 5x - 4 by 2x 2 -. Synthetic Division of Polynomials In the special case when the divisor is a linear factor of the form x a or x a, there is another, more efficient way to divide polynomials. This method is called synthetic division. It is called synthetic because it is a contrived shorthand way of dividing a polynomial by a linear factor. A detailed step-by-step procedure is given below for synthetic division. Let s divide x 4 x 3 2x 2 by x using synthetic division. Study Tip If (x a) is the divisor, then a is the number used in synthetic division. STEP Write the division in synthetic form. List the coefficients of the dividend. Remember to use 0 for a placeholder. The divisor is x. Since x 0 x is used. Coefficients of Dividend STEP 2 Bring down the first term () in the dividend. STEP 3 Multiply the by this leading coefficient (), and place the product up and to the right in the second column. Bring down the

6 4.3 Dividing Polynomials: Long Division and Synthetic Division 45 STEP 4 Add the values in the second column. STEP 5 Repeat Steps 3 and 4 until all columns are filled. ADD STEP 6 Identify the quotient by assigning powers of x in descending order, beginning with x n x 4 x 3. The last term is the remainder Remainder Quotient Coefficients x 3 2x 2 2x 4 f Study Tip Synthetic division can only be used when the divisor is of the form x a or x a. We know that the degree of the first term of the quotient is 3 because a fourth-degree polynomial was divided by a first-degree polynomial. Let s compare dividing x 4 x 3 2x 2 by x using both long division and synthetic division. Long Division x 3 2x 2 2x 4 x x 4 x 3 0x 2 2x 2 x 4 + x 3-2x 3 + 0x 2 -(-2x 3-2x 2 ) 2x 2-2x -(2x 2 + 2x) -4x + 2 -(-4x - 4) 6 Synthetic Division f x 3 2x 2 2x 4 Both long division and synthetic division yield the same answer. x 4 - x 3-2x + 2 x + = x 3-2x 2 + 2x x + EXAM PLE 6 Synthetic Division Use synthetic division to divide 3x 5 2x 3 x 2 7 by x 2. STEP Write the division in synthetic form. List the coefficients of the dividend. Remember to use 0 for a placeholder. The divisor of the original problem is x 2. If we set x 2 0 we find that x 2, so 2 is the divisor for synthetic division

7 46 CHAPTE R 4 Polynomial and Rational Functions STEP 2 Perform the synthetic division steps STEP 3 Identify the quotient and remainder f 3x 4 6x 3 0x 2 9x 38 3x 5-2x 3 + x 2-7 x + 2 = 3x 4-6x 3 + 0x 2-9x x + 2 Answer: 2x 2 + 2x YOU R TU R N Use synthetic division to divide 2x 3 x 3 by x. SECTION 4.3 SU M MARY Division of Polynomials Long division can always be used. Synthetic division is restricted to when the divisor is of the form x a or x a. Expressing Results Dividend Divisor = quotient + remainder divisor Dividend (quotient)(divisor) remainder When Remainder Is Zero Dividend (quotient)(divisor) Quotient and divisor are factors of the dividend. SECTION 4.3 EXE RCISES SKILLS In Exercises 30, divide the polynomials using long division. Use exact values. Express the answer in the form Q(x)?, r(x)?.. (2x 2 5x 3) (x 3) 2. (2x 2 5x 3) (x 3) 3. (x 2 5x 6) (x 2) 4. (2x 2 3x ) (x ) 5. (3x 2 9x 5) (x 2) 6. (x 2 4x 3) (x ) 7. (3x 2 3x 0) (x 5) 8. (3x 2 3x 0) (x 5) 9. (x 2 4) (x 4) 0. (x 2 9) (x 2). (9x 2 25) (3x 5) 2. (5x 2 3) (x ) 3. (4x 2 9) (2x 3) 4. (8x 3 27) (2x 3) 5. (x 20x 2 2x 3 2) (3x 2) 6. (2x 3 2 x 20x 2 ) (2x ) 7. (4x 3 2x 7) (2x ) 8. (6x 4 2x 2 5) ( 3x 2) 9. (4x3-2x 2 - x + 3), A 2 B 20. (2x x + 6x 2 ), Ax + 3 B 2. ( 2x 5 3x 4 2x 2 ) (x 3 3x 2 ) 22. ( 9x 6 7x 4 2x 3 5) (3x 4 2x )

8 4.3 Dividing Polynomials: Long Division and Synthetic Division 47 x 4 - x x + 7x 3 + 6x x 2 - x x 2 + x - 2-3x 4 + 7x 3-2x x (x 4 0.8x x x 0.044) (x 2.4x 0.49) 30. (x 5 2.8x 4.34x x x ) (x 2 0.6x 0.09) 2x 5-4x 3 + 3x x x 2 + 4x x 2-9 In Exercises 3 50, divide the polynomial by the linear factor with synthetic division. Indicate the quotient Q(x) and the remainder r(x). 3. (3x 2 7x 2) (x 2) 32. (2x 2 7x 5) (x 5) 33. (7x 2 3x 5) (x ) 34. (4x 2 x ) (x 2) 35. (3x 2 4x x 4 2x 3 4) (x 2) 36. (3x 2 4 x 3 ) (x ) 37. (x 4 ) (x ) 38. (x 4 9) (x 3) 39. (x 4 6) (x 2) 40. (x 4 8) (x 3) x 2 - x + ), Ax + 2 B x2 + ), Ax x 3 + 7x 2-4), Ax B 44. (3x 4 + x 3 + 2x - 3), Ax B 45. (2x 4 9x 3 9x 2 8x 8) (x.5) 46. (5x 3 x 2 6x 8) (x 0.8) x 7-8x 4 + 3x x 6 + 4x 5-2x x (x 6-49x 4-25x ), A5B 50. (x 6-4x 4-9x ), A3B 3 B In Exercises 5 60, divide the polynomials by either long division or synthetic division. 5. (6x 2 23x 7) (3x ) 52. (6x 2 x 2) (2x ) 53. (x 3 x 2 9x 9) (x ) 54. (x 3 2x 2 6x 2) (x 2) 55. (x 5 4x 3 2x 2 ) (x 2) 56. (x 4 x 2 3x 0) (x 5) 57. (x 4 25) (x 2 ) 58. (x 3 8) (x 2 2) 59. (x 7 ) (x ) 60. (x 6 27) (x 3) A P P L I C AT I O N S 6. Geometry. The area of a rectangle is 6x 4 4x 3 x 2 2x square feet. If the length of the rectangle is 2x 2 feet, what is the width of the rectangle? 62. Geometry. If the rectangle in Exercise 6 is the base of a rectangular box with volume 8x 5 8x 4 x 3 7x 2 5x cubic feet, what is the height of the box? 63. Travel. If a car travels a distance of x 3 60x 2 x 60 miles at an average speed of x 60 miles per hour, how long does the trip take? 64. Sports. If a quarterback throws a ball x 2 5x 50 yards in 5 x seconds, how fast is the football traveling?