Explanation Procedure used: Long Division of polynomials: “1. Arrange the terms of both the dividend and the divisor in descending powers of any variable. 2. Divide the first term in the dividend by the first term in the divisor. The result is the first term of the quotient. 3. Multiply every term in the divisor by the first term in the quotient. Write the resulting product beneath the dividend with like terms line up. 4. Subtract the product from the dividend. 5. Bring down the next term in the original dividend and write it next to the remainder to form a new dividend. 6. Use this new expression as the dividend and repeat this process until the remainder can no longer be divided. This will occur when the degree of the remainder (the highest exponent on a variable in the remainder) is less than the degree of the divisor”. Calculation: Divide the polynomial 2 x 5 − 8 x 4 + 2 x 3 + x 2 by 2 x 3 + 1 as follows. Step 1: The dividend 2 x 5 − 8 x 4 + 2 x 3 + x 2 is in descending order and write 0 as coefficient for the missing terms, so the dividend becomes 2 x 5 − 8 x 4 + 2 x 3 + x 2 + 0 x + 0 . 2 x 3 + 1 2 x 5 − 8 x 4 + 2 x 3 + x 2 + 0 x + 0 Step 2: Divide the first term of the dividend which is 2 x 5 by the first term of the divisor which is 2 x 3 . That is, 2 x 5 2 x 3 = x 2 . Align the like terms as follows. 2 x 3 + 1 x 2 2 x 5 − 8 x 4 + 2 x 3 + x 2 + 0 x + 0 Step 3: Multiply each term in 2 x 3 + 1 by x 2 and then align terms of product under like terms in dividend. x 2 ( 2 x 3 + 1 ) = 2 x 5 + x 2 That is, x 2 ( 2 x 3 + 1 ) = 2 x 5 + 0 x 4 + 0 x 3 + x 2 . 2 x 3 + 1 x 2 2 x 5 − 8 x 4 + 2 x 3 + x 2 + 0 x + 0 2 x 5 + 0 x 4 + 0 x 3 + x 2 _ Step 4: To subtract the polynomials 2 x 5 + 0 x 4 + 0 x 3 + x 2 from 2 x 5 − 8 x 4 + 2 x 3 + x 2 , it is needed to change the sign of the lower expression 2 x 5 + 0 x 4 + 0 x 3 + x 2 and then add. 32 x 3 + 1 x 2 2 x 5 − 8 x 4 + 2 x 3 + x 2 + 0 x + 0 2 x 5 + 0 x 4 + 0 x 3 + x 2 _ − 8 x 4 + 2 x 3 + 0 x 2 Step 5: Bring down the next term 0 x from the original dividend and add to form a new dividend. 2 x 3 + 1 x 2 2 x 5 − 8 x 4 + 2 x 3 + x 2 + 0 x + 0 2 x 5 + 0 x 4 + 0 x 3 + x 2 _ − 8 x 4 + 2 x 3 + 0 x 2 + 0 x Divide the first term of the dividend which is − 8 x 4 by the first term of the divisor which is 2 x 3 . That is, − 8 x 4 2 x 3 = − 4 x . Multiply each term in 2 x 3 + 1 by − 4 x and then align terms of product under like terms in dividend. − 4 x ( 2 x 3 + 1 ) = − 8 x 4 − 4 x That is, − 4 x ( 2 x 3 + 1 ) = − 8 x 4 + 0 x 3 + 0 x 2 − 4 x . 2 x 3 + 1 x 2 − 4 x 2 x 5 − 8 x 4 + 2 x 3 + x 2 + 0 x + 0 2 x 5 + 0 x 4 + 0 x 3 + x 2 _ − 8 x 4 + 2 x 3 + 0 x 2 + 0 x − 8 x 4 + 0 x 3 + 0 x 2 − 4 x _ To subtract the polynomials − 8 x 4 + 0 x 3 + 0 x 2 − 4 x from − 8 x 4 + 2 x 3 + 0 x 2 + 0 x , it is needed to change the sign of the lower expression − 8 x 4 + 0 x 3 + 0 x 2 − 4 x and then add. 2 x 3 + 1 x 2 − 4 x 2 x 5 − 8 x 4 + 2 x 3 + x 2 + 0 x + 0 2 x 5 + 0 x 4 + 0 x 3 + x 2 _ − 8 x 4 + 2 x 3 + 0 x 2 + 0 x − 8 x 4 + 0 x 3 + 0 x 2 − 4 x _ 2 x 3 + 0 x 2 + 4 x Bring down the next term 0 from the original dividend and add to form a new dividend

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ISBN: 9780137308286

Author: Blitzer

Publisher: RENT PEARS

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Chapter 3.3, Problem 16E

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To divide: The polynomial 2x5−8x4+2x3+x2 by 2x3+1.

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Chapter 3.3, Problem 15E

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To divide: The polynomial 2 x 5 − 8 x 4 + 2 x 3 + x 2 by 2 x 3 + 1 .

Solution Summary: The author explains how to divide the polynomial 2x5-8 x 4+2. The quotient and the divisor are in descending powers of any variable.

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To divide: The polynomial 2x5−8x4+2x3+x2 by 2x3+1.

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