(k^3-k^2-k-2)/(k-2)

Jasonbradley

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Jun 16, 2024

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Simplifying the Expression (k^3 - k^2 - k - 2) / (k - 2)

This article will guide you through simplifying the expression (k^3 - k^2 - k - 2) / (k - 2). We will use polynomial long division to achieve this.

Polynomial Long Division

1. Set up the division:

   • Write the dividend (k^3 - k^2 - k - 2) inside the division symbol.
   • Write the divisor (k - 2) outside the division symbol.

       ____________
   k - 2 | k^3 - k^2 - k - 2 
   

2. Divide the leading terms:

   • Divide the leading term of the dividend (k^3) by the leading term of the divisor (k).
   • This gives us k^2.
   • Write k^2 above the division symbol, aligning it with the k^3 term.

       k^2        
   k - 2 | k^3 - k^2 - k - 2 
   

3. Multiply the divisor by the quotient:

   • Multiply (k - 2) by k^2 to get k^3 - 2k^2.
   • Write this result below the dividend, aligning terms.

       k^2        
   k - 2 | k^3 - k^2 - k - 2 
          k^3 - 2k^2
   

4. Subtract:

   • Subtract (k^3 - 2k^2) from (k^3 - k^2 - k - 2).
   • This gives us k^2 - k - 2.

       k^2        
   k - 2 | k^3 - k^2 - k - 2 
          k^3 - 2k^2
          ---------
              k^2 - k - 2
   

5. Bring down the next term:

   • Bring down the next term from the dividend (-k).

       k^2        
   k - 2 | k^3 - k^2 - k - 2 
          k^3 - 2k^2
          ---------
              k^2 - k - 2
              - k
   

6. Repeat steps 2-5:

   • Divide the leading term of the new dividend (k^2) by the leading term of the divisor (k). This gives us k.
   • Write k above the division symbol, aligning it with the k^2 term.
   • Multiply (k - 2) by k to get k^2 - 2k.
   • Subtract (k^2 - 2k) from (k^2 - k - 2).
   • Bring down the next term (-2).

       k^2 + k     
   k - 2 | k^3 - k^2 - k - 2 
          k^3 - 2k^2
          ---------
              k^2 - k - 2
              - k
              ---------
                  k - 2 
   

7. Repeat steps 2-5:

   • Divide the leading term of the new dividend (k) by the leading term of the divisor (k). This gives us 1.
   • Write 1 above the division symbol, aligning it with the k term.
   • Multiply (k - 2) by 1 to get k - 2.
   • Subtract (k - 2) from (k - 2).
   • We are left with 0.

       k^2 + k + 1
   k - 2 | k^3 - k^2 - k - 2 
          k^3 - 2k^2
          ---------
              k^2 - k - 2
              - k
              ---------
                  k - 2 
                  k - 2
                  -----
                     0
   

Result

Therefore, the simplified expression is: (k^3 - k^2 - k - 2) / (k - 2) = k^2 + k + 1

Note: This result is valid for all values of k except for k = 2, as the original expression is undefined when k = 2 (division by zero).