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Dividing Polynomials Depends on the situation. Situation I: Polynomial Monomial Solution is to divide each term in the numerator by the monomial. For example:

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Dividing Polynomials Situation II: Polynomial Binomial Two methods of solution. Method 1 Long Division Remember the process for long division of numbers. For example:123 ÷ 3 3123 41 – 12 3 – 3 0

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Basics of Division Parts of Division 3123 41 divisor dividend quotient

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Dividing Polynomials Method 1 Long Division For example: x – 4x 2 – 11x + 28 Focus on the first term of the divisor. What does this term need to be multiplied by to equal the first term of the dividend? x Now multiply the entire binomial by this term. x 2 – 4x

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Dividing Polynomials Method 1 Long Division For example: x – 4x 2 – 11x + 28 Now subtract the terms. x x 2 – 4x – 7x+ 28 Now bring down the next term from the dividend. Repeat the process. So once again, focus on the first term of the binomial.

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Dividing Polynomials Method 1 Long Division For example: x – 4x 2 – 11x + 28 x – 7x + 28 What does x need to be multiplied by to get –7x? – 7 Once again multiply the entire binomial by this term. – 7x + 28 Now subtract the terms. x 2 – 4x 0 So the answer or quotient is (x – 7).

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Practice Divide the following using long division. Answer (Quotient): remainder

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Dividing Polynomials Method 2 Synthetic Division Step 1 Write the polynomial (dividend) in descending order of exponents and be sure to account for all powers of the variable. For example:(x 4 – 2x 3 + 2x – 1)(x + 1) -1 x 4 – 2x 3 + 0x 2 + 2x – 1 Step 2 Write coefficients of the dividend in order. 1 –2 0 2 –1

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Dividing Polynomials 1 –2 0 2 –1 Step 3 Write constant value r of divisor (x – r) to the left of the coefficients. In this case with the divisor (x + 1) has constant –1. Step 4 Bring down first coefficient. Multiply by constant and add result to second coefficient. Repeat process successively until last coefficient. –1 1 –3 3 3 –1 1 0 remainder x – ( – 1)

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Dividing Polynomials 1 –2 0 2 –1–1 1 –3 3 3 –1 1 0 Step 5 Write the quotient with the numbers along the bottom row as the coefficients of the powers of the variable in descending order. Start with the power that is one less than that of the dividend. In this case it is 3. Quotient: x 3 – 3x 2 + 3x – 1

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