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Polynomial Long Division: Examples and Guided Practice

Learn how to divide polynomials using long division with step-by-step examples and guided practice.

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Polynomial Long Division: Examples and Guided Practice

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  1. EXAMPLE 1 Use polynomial long division Divide f (x) = 3x4 – 5x3 + 4x – 6 byx2 – 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.

  2. quotient ) x2 – 3x + 5 3x4 – 5x3 + 0x2 + 4x – 6 3x4 – 9x3 + 15x2 4x3– 12x2 + 20x –3x2 + 9x – 15 remainder EXAMPLE 1 Use polynomial long division 3x2+ 4x – 3 Multiply divisor by 3x4/x2 = 3x2 Subtract. Bring down next term. 4x3–15x2+ 4x Multiply divisor by4x3/x2 = 4x Subtract. Bring down next term. – 3x2 – 16x – 6 Multiply divisor by– 3x2/x2 = – 3 – 25x + 9

  3. 3x4 – 5x3 + 4x – 6 – 25x + 9 ANSWER = 3x2 + 4x – 3 + x2 – 3x + 5 x2 – 3x + 5 = 3x4–5x3+ 4x –6 EXAMPLE 1 Use polynomial long division 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

  4. quotient ) x – 2 x3 + 5x2 – 7x + 2 x3 – 2x2 7x2– 14x 7x – 14 remainder x3 + 5x2– 7x +2 16 ANSWER = x2 + 7x + 7 + x – 2 x – 2 EXAMPLE 2 Use polynomial long division with a linear divisor Divide f(x) = x3 + 5x2 – 7x + 2 by x – 2. x2+ 7x + 7 Multiply divisor byx3/x = x2. 7x2 – 7x Subtract. Multiply divisor by7x2/x = 7x. 7x + 2 Subtract. Multiply divisor by7x/x = 7. 16

  5. for Examples 1 and 2 GUIDED PRACTICE Divide using polynomial long division. 1. (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.

  6. quotient ) x2 + 2x – 1 2x4 + x3 + 0x2 + x – 1 2x4 – 4x3 – 2x2 – 3x3– 6x2 + 3x 8x2 –16x – 8 remainder for Examples 1 and 2 GUIDED PRACTICE 2x2 – 3x + 8 Multiply divisor by 2x4/x2 = –2x2. Subtract. Bring down next term. 3x3 – 2x2 + x Multiply divisor by –3x3/x2 = –3. Subtract. Bring down next term. 8x2 – 2x – 1 Multiply divisor by4x2/x2 = 8. – 18x + 7

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

  8. for Examples 1 and 2 GUIDED PRACTICE 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.

  9. quotient ) x + 2 x3 – x2 + 4x – 10 x3 + 2x2 – 3x2– 6x 10x + 20 remainder for Examples 1 and 2 GUIDED PRACTICE x2 – 3x + 10 Multiply divisor byx3/x = x2. Subtract. Bring down next term. –3x2 + 4x Multiply divisor by –3x2/x= –3x. Subtract. Bring down next term. 10x – 1 Multiply divisor by10x/x = 10. – 30

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

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