(x^3-1)/(x^2-1) Long Division

Jasonbradley

4 min read

·

Jun 17, 2024

29

Share

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

1. 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 
   

2. 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

1. 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 
   

2. 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     
   

3. 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 
   

4. 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 
   

5. 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
   

6. 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.