Section 3-1.3

# Section 3-1.3 - x r x q x d x f" f = dividend d =...

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3.3 – Dividing Polynomials; Remainder and Factor Theorems The polynomials that we solved thus far have been factorable using basic factoring rules. In order to work with polynomials that do not factor as easily, we need to first review division of polynomials. Long Division of Polynomials 1. Arrange the terms in the dividend and the divisor in descending powers of the variable. Use zero placeholders for any missing terms. 2. Divide the first term in the dividend by the first term in the divisor. This result becomes the first term in the quotient. 3. Multiply this new term by each term in the divisor and create a new row. 4. Subtract rows. 5. Bring down the remaining terms in the dividend and repeat from Step 2. 6. The process will stop once the degree of the remaining polynomial on the inside is less than the degree of the divisor. You can write your final answer in one of two ways: the quotient plus the remainder as a fraction of the divisor OR use the division algorithm. Division Algorithm

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Unformatted text preview: ) ( ) ( ) ( ) ( x r x q x d x f ! " # f = dividend d = divisor q = quotient r = remainder ! Example 1 Divide 7 11 3 2 2 3 ! \$ \$ x x x by 3 \$ x . " ! Example 2 Divide 10 7 3 2 2 3 \$ \$ ! x x x by x x 2 2 \$ . " If the divisor is of the form c x \$ where c is a constant (positive or negative), then the long division process can be replaced by synthetic division . In synthetic division, only the coefficients are used. The quotient will always be one degree less than the dividend. ! Example 3 Use synthetic division to divide 6 7 3 \$ \$ x x by 2 ! x . " Remainder Theorem If the polynomial ) ( x f is divided by c x \$ , then the remainder is equal to ) ( c f . ! Example 4 Given 3 5 4 3 ) ( 2 3 ! \$ ! # x x x x f , use the Remainder Theorem to find ) 4 ( \$ f . " Factor Theorem If ) ( # c f , then c x \$ is a factor of ) ( x f . If c x \$ is a factor of ) ( x f , then ) ( # c f . ! Example 5 Solve the equation 2 3 14 15 2 3 # \$ \$ ! x x x given that -1 is a zero of the function. "...
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## This note was uploaded on 10/18/2011 for the course MAC 1105 taught by Professor Staff during the Spring '08 term at University of Central Florida.

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Section 3-1.3 - x r x q x d x f" f = dividend d =...

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