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(2x^3-5x^2+3x+7)/(x-2) Quizizz

5 min read Jun 16, 2024
(2x^3-5x^2+3x+7)/(x-2) Quizizz

Polynomial Long Division: (2x^3-5x^2+3x+7)/(x-2)

This article will guide you through the process of dividing the polynomial 2x^3 - 5x^2 + 3x + 7 by x - 2 using polynomial long division. This is a common technique used in algebra and is often found in quizzes like those on Quizizz.

Understanding Polynomial Long Division

Polynomial long division is similar to the long division you learned with numbers. We'll systematically divide the dividend (2x^3 - 5x^2 + 3x + 7) by the divisor (x - 2) to find the quotient and remainder.

Step-by-Step Solution

  1. Set up the division:

         _______
x - 2 | 2x^3 - 5x^2 + 3x + 7

  2. Divide the leading terms:

    • Divide the leading term of the dividend (2x^3) by the leading term of the divisor (x): 2x^3 / x = 2x^2.
    • Write the result (2x^2) above the dividend.
         2x^2 _______
x - 2 | 2x^3 - 5x^2 + 3x + 7

  3. Multiply the divisor by the result:

    • Multiply the divisor (x - 2) by 2x^2: (x - 2) * 2x^2 = 2x^3 - 4x^2.
    • Write the result below the dividend.
         2x^2 _______
x - 2 | 2x^3 - 5x^2 + 3x + 7 
       2x^3 - 4x^2

  4. Subtract:

    • Subtract the result from the dividend. Remember to change the signs of the terms in the second row before subtracting.
         2x^2 _______
x - 2 | 2x^3 - 5x^2 + 3x + 7 
       2x^3 - 4x^2
       ---------
           -x^2 + 3x

  5. Bring down the next term:

    • Bring down the next term from the dividend (+3x).
         2x^2 _______
x - 2 | 2x^3 - 5x^2 + 3x + 7 
       2x^3 - 4x^2
       ---------
           -x^2 + 3x + 7

  6. Repeat steps 2-5:

    • Divide the new leading term (-x^2) by the leading term of the divisor (x): -x^2 / x = -x.
    • Write the result (-x) above the dividend.
    • Multiply the divisor (x - 2) by -x: (x - 2) * -x = -x^2 + 2x.
    • Subtract the result from the previous row.
    • Bring down the next term (+7).
         2x^2 - x _______
x - 2 | 2x^3 - 5x^2 + 3x + 7 
       2x^3 - 4x^2
       ---------
           -x^2 + 3x + 7
           -x^2 + 2x
           ---------
               x + 7

  7. Repeat steps 2-5 again:

    • Divide the new leading term (x) by the leading term of the divisor (x): x / x = 1.
    • Write the result (1) above the dividend.
    • Multiply the divisor (x - 2) by 1: (x - 2) * 1 = x - 2.
    • Subtract the result from the previous row.
         2x^2 - x + 1 _______
x - 2 | 2x^3 - 5x^2 + 3x + 7 
       2x^3 - 4x^2
       ---------
           -x^2 + 3x + 7
           -x^2 + 2x
           ---------
               x + 7 
               x - 2
               ---------
                   9

  8. Result:

    • The quotient is 2x^2 - x + 1.
    • The remainder is 9.

    Therefore, (2x^3 - 5x^2 + 3x + 7) / (x - 2) = 2x^2 - x + 1 + 9/(x - 2).

Key Takeaways

  • Polynomial long division is a systematic way to divide polynomials.
  • It involves repeated steps of dividing, multiplying, subtracting, and bringing down terms.
  • The result includes a quotient and a remainder.

Now you can confidently tackle similar problems on Quizizz and understand the underlying principles of polynomial long division.

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