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

Jasonbradley

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Jun 17, 2024

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

This article will demonstrate how to perform long division on the polynomial expression (x^3+x^2+x+2)/(x^2-1).

Setting up the Division

1. Write the dividend (numerator) inside the division symbol:

       __________
   x^2-1 | x^3 + x^2 + x + 2
   

2. Write the divisor (denominator) outside the division symbol:

       __________
   x^2-1 | x^3 + x^2 + x + 2
   

Performing the Division

1. Divide the leading term of the dividend (x^3) by the leading term of the divisor (x^2):

   • x^3 / x^2 = x
   • Write the quotient (x) above the x^2 term in the dividend.

       x     
       __________
   x^2-1 | x^3 + x^2 + x + 2
   

2. Multiply the divisor (x^2-1) by the quotient (x):

   • (x^2-1) * x = x^3 - x
   • Write the result below the dividend, aligning terms with the same powers:

       x     
       __________
   x^2-1 | x^3 + x^2 + x + 2
           x^3 - x
   

3. Subtract the result from the dividend:

   • (x^3 + x^2 + x + 2) - (x^3 - x) = x^2 + 2x + 2

       x     
       __________
   x^2-1 | x^3 + x^2 + x + 2
           x^3 - x
           -------
               x^2 + 2x + 2
   

4. Bring down the next term from the dividend (2):

       x     
       __________
   x^2-1 | x^3 + x^2 + x + 2
           x^3 - x
           -------
               x^2 + 2x + 2
   

5. Repeat steps 1-4 with the new dividend (x^2 + 2x + 2):

   • Divide the leading term of the new dividend (x^2) by the leading term of the divisor (x^2): x^2 / x^2 = 1
   • Write the quotient (1) next to the x in the quotient:

       x + 1 
       __________
   x^2-1 | x^3 + x^2 + x + 2
           x^3 - x
           -------
               x^2 + 2x + 2
   

   • Multiply the divisor (x^2-1) by the new quotient (1):
   • (x^2-1) * 1 = x^2 - 1
   • Write the result below the new dividend, aligning terms with the same powers:

       x + 1 
       __________
   x^2-1 | x^3 + x^2 + x + 2
           x^3 - x
           -------
               x^2 + 2x + 2
               x^2 - 1 
   

   • Subtract the result from the new dividend:
   • (x^2 + 2x + 2) - (x^2 - 1) = 2x + 3

       x + 1 
       __________
   x^2-1 | x^3 + x^2 + x + 2
           x^3 - x
           -------
               x^2 + 2x + 2
               x^2 - 1 
               -------
                   2x + 3 
   

6. The degree of the new dividend (2x + 3) is less than the degree of the divisor (x^2-1). This means we stop here.

The Result

The result of the long division is:

(x^3 + x^2 + x + 2) / (x^2 - 1) = x + 1 + (2x + 3)/(x^2 - 1)

This can be rewritten as a mixed number:

(x^3 + x^2 + x + 2) / (x^2 - 1) = x + 1 + (2x + 3)/(x^2 - 1)