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Long and Synthetic Division of Polynomials Section 2-3

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2 Objectives I can use long division to divide two polynomials I can use synthetic division to divide a polynomial by a binomial (x – r)

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3 Dividing Numbers When you divide a number by another number and there is no remainder: Then the divisor is a factor!! Also the quotient becomes another factor!!! Dividend Divisor Quotient

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4 Dividing Polynomials Long division of polynomials is similar to long division of whole numbers. dividend = (quotient divisor) + remainder The result is written in the form: quotient + When you divide two polynomials you can check the answer using the following:

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5 + 2 Dividing Polynomials Example: Divide x 2 + 3x – 2 by x + 1 and check the answer. x x 2 + x 2x2x– 2 2x + 2 – 4– 4 remainder 1. 2. 3. 4. 5. 6. Answer: x + 2 + – 4– 4 Dividing Polynomials

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6 Example: Divide & Check Example: Divide 4x + 2x 3 – 1 by 2x – 2 and check the answer. Write the terms of the dividend in descending order. 1. x2x2 2. 2x 3 – 2x 2 3. 2x22x2 + 4x 4. + x 5. 2x 2 – 2x 6. 6x6x – 1 7. + 3 8. 6x – 6 9. 5 Answer: x 2 + x + 3 5 Since there is no x 2 term in the dividend, add 0x 2 as a placeholder.

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7 Example: Division With Zero Remainder x x 2 – 2x – 3x + 6 – 3 – 3x + 6 0 Answer: x – 3 with no remainder. Example: Divide x 2 – 5x + 6 by x – 2.

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8 Dividing by Synthetic Division Synthetic Division is a method to divide any polynomial by a binomial. The steps must be followed exactly in order or you will not get the correct end result The following slide shows the steps for one complete problem.

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9 Find: (6x 3 - 19x 2 + x + 6) (x-3) Step 1: Rewrite the dividend with all terms. If a term is missing, insert a zero for that term. Bring down the coefficients from the dividend and make a row. Next identify the divisor. It must be in the format (x-r). Bring down r and put in a box on the left. Draw a line. Bring down 1 st coefficient under the line. Multiply it by “r” and add to next column. Then repeat. New row of numbers are the coefficients of the quotient starting with one power less. 6x 3 – 19x 2 + 1x + 6 6 -19 1 6 3 6 18 -3 -2 -6 0 6x 2 – 1x – 2 (No remainder)

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10 Find: (4x 4 - 5x 2 + 2x + 4) (x+1) Step 1: Rewrite the dividend with all terms. If a term is missing, insert a zero for that term. Bring down the coefficients from the dividend and make a row. Next identify the divisor. It must be in the format (x-r). Bring down r and put in a box on the left. Draw a line. Bring down 1 st coefficient under the line. Multiply it by “r” and add to next column. Then repeat. New row of numbers are the coefficients of the quotient starting with one power less. 4x 4 + 0x 3 – 5x 2 + 2x + 4 4 0 -5 2 4 4 -4 4 1 3 -3 1

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11 16 Synthetic Division Synthetic division is a shorter method of dividing polynomials. This method can be used only when the divisor is of the form x – a. It uses the coefficients of each term in the dividend. Example: Divide 3x 2 + 2x – 1 by x – 2 using synthetic division. 3 2 – 1 2 Since the divisor is x – 2, a = 2. 3 1. Bring down 3 2. (2 3) = 6 6 815 3. (2 + 6) = 8 4. (2 8) = 16 5. (–1 + 16) = 15 coefficients of quotient remainder value of a coefficients of the dividend 3x + 8Answer: 15

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12 Homework WS 4-1

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