(5x^3-6x^2+8)/(x-4)

Jasonbradley

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Jun 16, 2024

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Dividing Polynomials: A Step-by-Step Guide

This article will guide you through the process of dividing the polynomial 5x³ - 6x² + 8 by (x - 4) using polynomial long division.

Understanding Polynomial Long Division

Polynomial long division is similar to the long division we learned in elementary school, but instead of dealing with numbers, we work with polynomials. The goal is to find the quotient (the result of the division) and the remainder.

Step-by-Step Solution

1. Set up the division problem:

       ___________
   x - 4 | 5x³ - 6x² + 0x + 8 
   

   • Notice that we added a "0x" term to the dividend to ensure all terms are present. This helps us maintain the correct place values during the division process.

2. Divide the leading terms:

   • Divide 5x³ by x, which gives 5x². Write this above the line, aligned with the x² term.

       5x² _______
   x - 4 | 5x³ - 6x² + 0x + 8 
   

3. Multiply the quotient (5x²) by the divisor (x - 4):

   • This gives 5x³ - 20x². Write this below the dividend.

       5x² _______
   x - 4 | 5x³ - 6x² + 0x + 8 
           5x³ - 20x² 
   

4. Subtract the results:

   • Change the signs of the terms in the second row and add.

       5x² _______
   x - 4 | 5x³ - 6x² + 0x + 8 
           5x³ - 20x² 
           ----------
                  14x² + 0x 
   

5. Bring down the next term:

   • Bring down the next term from the dividend, which is 0x.

       5x² _______
   x - 4 | 5x³ - 6x² + 0x + 8 
           5x³ - 20x² 
           ----------
                  14x² + 0x 
   

6. Repeat steps 2-5:

   • Divide the leading term of the new dividend, 14x², by x, which gives 14x.
   • Multiply 14x by (x - 4) to get 14x² - 56x.
   • Subtract this from the current dividend.
   • Bring down the next term, 8.

       5x² + 14x ______
   x - 4 | 5x³ - 6x² + 0x + 8 
           5x³ - 20x² 
           ----------
                  14x² + 0x
                  14x² - 56x
                  ---------
                          56x + 8
   

7. Repeat steps 2-5 again:

   • Divide 56x by x, which gives 56.
   • Multiply 56 by (x - 4) to get 56x - 224.
   • Subtract this from the current dividend.

       5x² + 14x + 56 ______
   x - 4 | 5x³ - 6x² + 0x + 8 
           5x³ - 20x² 
           ----------
                  14x² + 0x
                  14x² - 56x
                  ---------
                          56x + 8
                          56x - 224
                          -------
                                232 
   

8. The result:

   • We have a remainder of 232.
   • Therefore, the quotient is 5x² + 14x + 56 and the remainder is 232.

Final Answer

The result of dividing 5x³ - 6x² + 8 by (x - 4) can be expressed as:

5x² + 14x + 56 + 232/(x - 4)