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

Jasonbradley

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Jun 16, 2024

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Long Division of Polynomials: (w^3 - 2w^2 - 2w + 1) / (w - 1)

Long division of polynomials is a method used to divide one polynomial by another, similar to the long division of numbers. This process involves a series of steps to find the quotient and remainder.

Let's illustrate this with the example of dividing (w^3 - 2w^2 - 2w + 1) by (w - 1):

Step 1: Set up the division.

          _______
w - 1 | w^3 - 2w^2 - 2w + 1 

Step 2: Divide the leading terms.

• The leading term of the divisor (w - 1) is 'w'.
• The leading term of the dividend (w^3 - 2w^2 - 2w + 1) is 'w^3'.
• Divide 'w^3' by 'w' to get 'w^2'. This is the first term of the quotient.

          w^2 _______
w - 1 | w^3 - 2w^2 - 2w + 1 

Step 3: Multiply the quotient term by the divisor.

• Multiply 'w^2' by (w - 1) to get 'w^3 - w^2'.

          w^2 _______
w - 1 | w^3 - 2w^2 - 2w + 1 
          w^3 - w^2

Step 4: Subtract the result from the dividend.

• Subtract 'w^3 - w^2' from the dividend.

          w^2 _______
w - 1 | w^3 - 2w^2 - 2w + 1 
          w^3 - w^2
          -------
               -w^2 - 2w 

Step 5: Bring down the next term.

• Bring down the next term of the dividend, which is '-2w'.

          w^2 _______
w - 1 | w^3 - 2w^2 - 2w + 1 
          w^3 - w^2
          -------
               -w^2 - 2w + 1

Step 6: Repeat steps 2-5.

• Now, the leading term of the new dividend is '-w^2'.
• Divide '-w^2' by 'w' to get '-w'. This is the next term of the quotient.

          w^2 - w _______
w - 1 | w^3 - 2w^2 - 2w + 1 
          w^3 - w^2
          -------
               -w^2 - 2w + 1
               -w^2 + w 

• Multiply '-w' by (w - 1) to get '-w^2 + w'.
• Subtract this result from the previous line.

          w^2 - w _______
w - 1 | w^3 - 2w^2 - 2w + 1 
          w^3 - w^2
          -------
               -w^2 - 2w + 1
               -w^2 + w 
               -------
                      -3w + 1

• Bring down the next term, '1'.

          w^2 - w _______
w - 1 | w^3 - 2w^2 - 2w + 1 
          w^3 - w^2
          -------
               -w^2 - 2w + 1
               -w^2 + w 
               -------
                      -3w + 1

Step 7: Repeat steps 2-5 again.

• Divide '-3w' by 'w' to get '-3'. This is the final term of the quotient.

          w^2 - w - 3 _______
w - 1 | w^3 - 2w^2 - 2w + 1 
          w^3 - w^2
          -------
               -w^2 - 2w + 1
               -w^2 + w 
               -------
                      -3w + 1
                      -3w + 3

• Multiply '-3' by (w - 1) to get '-3w + 3'.
• Subtract this result from the previous line.

          w^2 - w - 3 _______
w - 1 | w^3 - 2w^2 - 2w + 1 
          w^3 - w^2
          -------
               -w^2 - 2w + 1
               -w^2 + w 
               -------
                      -3w + 1
                      -3w + 3
                      -------
                          -2 

Step 8: Identify the quotient and remainder.

• The quotient is the polynomial obtained on top: w^2 - w - 3.
• The remainder is the final result after the last subtraction: -2.

Therefore, the result of dividing (w^3 - 2w^2 - 2w + 1) by (w - 1) can be expressed as:

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