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EXAMPLE 1 Use polynomial long division Divide f (x) = 3x4 – 5x3 + 4x – 6 by x2 – 3x + 5. SOLUTION Write polynomial division in the same format you use when dividing numbers. Include a “0” as the coefficient of x2 in the dividend. At each stage, divide the term with the highest power in what is left of the dividend by the first term of the divisor. This gives the next term of the quotient.

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**) EXAMPLE 1 Use polynomial long division 3x2 + 4x – 3 x2 – 3x + 5**

quotient x2 – 3x + 5 3x4 – 5x3 + 0x2 + 4x – 6 ) Multiply divisor by 3x4/x2 = 3x2 3x4 – 9x3 + 15x2 Subtract. Bring down next term. 4x3 – 15x x 4x3 – 12x2 + 20x Multiply divisor by 4x3/x2 = 4x Subtract. Bring down next term. – 3x2 – 16x – 6 –3x2 + 9x – 15 Multiply divisor by – 3x2/x2 = – 3 – 25x + 9 remainder

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EXAMPLE 1 Use polynomial long division 3x4 – 5x3 + 4x – 6 x2 – 3x + 5 = 3x2 + 4x – 3 + – 25x + 9 ANSWER CHECK You can check the result of a division problem by multiplying the quotient by the divisor and adding the remainder. The result should be the dividend. (3x2 + 4x – 3)(x2 – 3x + 5) + (– 25x + 9) = 3x2(x2 – 3x + 5) + 4x(x2 – 3x + 5) – 3(x2 – 3x + 5) – 25x + 9 = 3x4 – 9x3 + 15x2 + 4x3 – 12x2 + 20x – 3x2 + 9x – 15 – 25x + 9 = 3x4 – 5x3 + 4x – 6

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**) EXAMPLE 2 Use polynomial long division with a linear divisor**

Divide f (x) = x3 + 5x2 – 7x + 2 by x – 2. x2 + 7x quotient x – 2 x3 + 5x2 – 7x ) x3 – 2x2 Multiply divisor by x3/x = x2. 7x2 – 7x Subtract. 7x2 – 14x Multiply divisor by 7x2/x = 7x. 7x + 2 Subtract. 7x – 14 Multiply divisor by 7x/x = 7. 16 remainder ANSWER x3 + 5x2 – 7x +2 x – 2 = x2 + 7x + 7 + 16

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GUIDED PRACTICE for Examples 1 and 2 Divide using polynomial long division. (2x4 + x3 + x – 1) (x2 + 2x – 1) SOLUTION Write polynomial division in the same format you use when dividing numbers. Include a “0” as the coefficient of x2 in the dividend. At each stage, divide the term with the highest power in what is left of the dividend by the first term of the divisor. This gives the next term of the quotient.

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**) GUIDED PRACTICE for Examples 1 and 2 2x2 – 3x + 8 x2 + 2x – 1**

quotient x2 + 2x – 1 2x4 + x3 + 0x2 + x – 1 ) Multiply divisor by 2x4/x2 = –2x2. 2x4 – 4x3 – 2x2 Subtract. Bring down next term. 3x3 – 2x2 + x – 3x3 – 6x2 + 3x Multiply divisor by –3x3/x2 = –3. Subtract. Bring down next term. 8x2 – 2x – 1 8x2 –16x – 8 Multiply divisor by 4x2/x2 = 8. – 18x + 7 remainder

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GUIDED PRACTICE for Examples 1 and 2 2x4 + 5x3 + x – 1 x2 + 2x – 1 = (2x2 – 3x + 8)+ – 18x + 7 ANSWER

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GUIDED PRACTICE for Examples 1 and 2 2. (x3 – x2 + 4x – 10) (x + 2) SOLUTION Write polynomial division in the same format you use when dividing numbers. Include a “0” as the coefficient of x2 in the dividend. At each stage, divide the term with the highest power in what is left of the dividend by the first term of the divisor. This gives the next term of the quotient.

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**) GUIDED PRACTICE for Examples 1 and 2 x2 – 3x + 10 x + 2**

quotient x + 2 x3 – x2 + 4x – 10 ) Multiply divisor by x3/x = x2. x3 + 2x2 Subtract. Bring down next term. –3x2 + 4x – 3x2 – 6x Multiply divisor by –3x2/x = –3x. Subtract. Bring down next term. 10x – 1 10x + 20 Multiply divisor by 10x/x = 10. – 30 remainder

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GUIDED PRACTICE for Examples 1 and 2 x3 – x2 +4x – 10 x + 2 = (x2 – 3x +10)+ – 30 ANSWER

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