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

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

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

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

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

Repeat steps 14:
 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

Repeat steps 14:
 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.