(2x^3-3x^2-18x-8)/(x-4)

Jasonbradley

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Jun 16, 2024

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Polynomial Division: (2x^3 - 3x^2 - 18x - 8) / (x - 4)

This article will guide you through the process of dividing the polynomial (2x^3 - 3x^2 - 18x - 8) by (x - 4) using polynomial long division.

Steps for Polynomial Long Division:

1. Set up the division: Write the dividend (2x^3 - 3x^2 - 18x - 8) inside the division symbol and the divisor (x - 4) outside.

         _________
   x - 4 | 2x^3 - 3x^2 - 18x - 8
   

2. Divide the leading terms: Divide the leading term of the dividend (2x^3) by the leading term of the divisor (x). This gives us 2x^2. Write this term above the line.

         2x^2      
   x - 4 | 2x^3 - 3x^2 - 18x - 8
   

3. Multiply the divisor: Multiply the term we just wrote (2x^2) by the entire divisor (x - 4). This gives us 2x^3 - 8x^2.

         2x^2      
   x - 4 | 2x^3 - 3x^2 - 18x - 8
         2x^3 - 8x^2
   

4. Subtract: Subtract the result (2x^3 - 8x^2) from the dividend. Remember to change the signs of the terms you are subtracting.

         2x^2      
   x - 4 | 2x^3 - 3x^2 - 18x - 8
         2x^3 - 8x^2
         -------------
                 5x^2 
   

5. Bring down the next term: Bring down the next term from the dividend (-18x).

         2x^2      
   x - 4 | 2x^3 - 3x^2 - 18x - 8
         2x^3 - 8x^2
         -------------
                 5x^2 - 18x 
   

6. Repeat the process: Divide the leading term of the new dividend (5x^2) by the leading term of the divisor (x). This gives us 5x. Write this term above the line.

         2x^2 + 5x    
   x - 4 | 2x^3 - 3x^2 - 18x - 8
         2x^3 - 8x^2
         -------------
                 5x^2 - 18x 
   

7. Multiply and subtract: Multiply 5x by (x - 4) to get 5x^2 - 20x. Subtract this from the current line.

         2x^2 + 5x    
   x - 4 | 2x^3 - 3x^2 - 18x - 8
         2x^3 - 8x^2
         -------------
                 5x^2 - 18x 
                 5x^2 - 20x
                 ---------
                         2x 
   

8. Bring down the next term: Bring down the last term of the dividend (-8).

         2x^2 + 5x    
   x - 4 | 2x^3 - 3x^2 - 18x - 8
         2x^3 - 8x^2
         -------------
                 5x^2 - 18x 
                 5x^2 - 20x
                 ---------
                         2x - 8
   

9. Repeat the process again: Divide the leading term of the new dividend (2x) by the leading term of the divisor (x). This gives us 2. Write this term above the line.

         2x^2 + 5x + 2
   x - 4 | 2x^3 - 3x^2 - 18x - 8
         2x^3 - 8x^2
         -------------
                 5x^2 - 18x 
                 5x^2 - 20x
                 ---------
                         2x - 8
                         2x - 8
                         -------
                               0
   

10. The result: Since we get a remainder of 0, we have successfully divided the polynomial.

Conclusion:

Therefore, (2x^3 - 3x^2 - 18x - 8) / (x - 4) = 2x^2 + 5x + 2. This means that (x - 4) is a factor of the polynomial (2x^3 - 3x^2 - 18x - 8).