Performing Long Division with (x^3  1) / (x^2  1)
This article demonstrates how to perform long division with the expression (x^3  1) / (x^2  1).
Setting up the Division

Write the dividend and divisor:
 The dividend is the expression being divided: (x^3  1)
 The divisor is the expression dividing the dividend: (x^2  1)
x^2  1  x^3  1

Focus on the leading terms:
 The leading term of the divisor (x^2) should be considered when deciding what to multiply by.
 The leading term of the dividend (x^3) is the focus for this step.
Performing the Division

Determine the first term of the quotient:
 Ask yourself: "What do I need to multiply x^2 by to get x^3?"
 The answer is x.
 Write "x" above the x^3 term in the dividend.
x x^2  1  x^3  1

Multiply the divisor by the quotient term:
 Multiply (x^2  1) by x: x * (x^2  1) = x^3  x
x x^2  1  x^3  1 x^3  x

Subtract the product from the dividend:
 Subtract (x^3  x) from (x^3  1). Remember to distribute the negative sign.
x x^2  1  x^3  1 x^3  x  x  1

Bring down the next term:
 Bring down the "1" term from the dividend.
x x^2  1  x^3  1 x^3  x  x  1

Repeat the process:
 Now focus on the new leading term (x).
 Ask: "What do I need to multiply x^2 by to get x?"
 The answer is 1/x.
 Write "1/x" above the "1" term in the dividend.
x + 1/x x^2  1  x^3  1 x^3  x  x  1 x  1/x

Multiply and subtract:
 Multiply (x^2  1) by 1/x: (1/x) * (x^2  1) = x  1/x
 Subtract (x  1/x) from (x  1).
x + 1/x x^2  1  x^3  1 x^3  x  x  1 x  1/x  1 + 1/x
The Result
We have a remainder of (1 + 1/x). Therefore, the long division result is:
x + 1/x + (1 + 1/x) / (x^2  1)
This result can be simplified further, but the process above demonstrates the mechanics of long division with polynomial expressions.