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(x^3-27)/(x-3) Long Division

4 min read Jun 17, 2024
(x^3-27)/(x-3) Long Division

Long Division of (x^3 - 27) / (x - 3)

Long division is a useful method for dividing polynomials. Let's work through the steps of dividing (x^3 - 27) by (x - 3).

Setting up the Division

  1. Write the problem:
         ___________
x - 3 | x^3 + 0x^2 + 0x - 27
    We include the placeholders (0x^2 and 0x) for the missing terms in the dividend (x^3 - 27) to maintain proper alignment.

Steps of the Division

  1. Divide the leading terms:

    • Divide x^3 (the leading term of the dividend) by x (the leading term of the divisor), resulting in x^2.
         x^2 ______
x - 3 | x^3 + 0x^2 + 0x - 27

  2. Multiply the quotient term by the divisor:

    • Multiply x^2 by (x - 3), resulting in x^3 - 3x^2.
         x^2 ______
x - 3 | x^3 + 0x^2 + 0x - 27
        x^3 - 3x^2

  3. Subtract:

    • Subtract (x^3 - 3x^2) from the dividend.
         x^2 ______
x - 3 | x^3 + 0x^2 + 0x - 27
        x^3 - 3x^2 
        ---------
               3x^2 + 0x

  4. Bring down the next term:

    • Bring down the next term of the dividend (0x).
         x^2 ______
x - 3 | x^3 + 0x^2 + 0x - 27
        x^3 - 3x^2 
        ---------
               3x^2 + 0x

  5. Repeat steps 1-4:

    • Divide 3x^2 by x, resulting in 3x.
    • Multiply 3x by (x - 3), resulting in 3x^2 - 9x.
    • Subtract (3x^2 - 9x) from the previous result.
    • Bring down the next term (-27).
         x^2 + 3x ______
x - 3 | x^3 + 0x^2 + 0x - 27
        x^3 - 3x^2 
        ---------
               3x^2 + 0x 
               3x^2 - 9x
               ---------
                      9x - 27

  6. Repeat steps 1-4:

    • Divide 9x by x, resulting in 9.
    • Multiply 9 by (x - 3), resulting in 9x - 27.
    • Subtract (9x - 27) from the previous result. The remainder is 0.
         x^2 + 3x + 9 
x - 3 | x^3 + 0x^2 + 0x - 27
        x^3 - 3x^2 
        ---------
               3x^2 + 0x 
               3x^2 - 9x
               ---------
                      9x - 27 
                      9x - 27
                      ---------
                           0

Result

Therefore, (x^3 - 27) / (x - 3) = x^2 + 3x + 9.

Important Notes

  • The remainder is 0, indicating that (x - 3) is a factor of (x^3 - 27).
  • The degree of the quotient is one less than the degree of the dividend.
  • Long division is a powerful tool for dividing polynomials, and understanding this process can help you solve various algebraic problems.

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