(x^4-5x^3+10x^2-3x-2)/(x-1)

Jasonbradley

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Jun 17, 2024

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Polynomial Long Division: (x^4-5x^3+10x^2-3x-2)/(x-1)

This article will demonstrate how to divide the polynomial x^4-5x^3+10x^2-3x-2 by x-1 using polynomial long division.

Steps for Polynomial Long Division:

1. Set up the division problem. Write the dividend (x^4-5x^3+10x^2-3x-2) inside the division symbol and the divisor (x-1) outside the division symbol.

       ____________
   x-1 | x^4 - 5x^3 + 10x^2 - 3x - 2 
   

2. Divide the leading term of the dividend by the leading term of the divisor. In this case, x^4 divided by x equals x^3. Write this quotient above the x^3 term of the dividend.

       x^3 ________
   x-1 | x^4 - 5x^3 + 10x^2 - 3x - 2 
   

3. Multiply the divisor by the quotient term. Multiply (x-1) by x^3, which gives x^4 - x^3. Write this result below the dividend.

       x^3 ________
   x-1 | x^4 - 5x^3 + 10x^2 - 3x - 2 
         x^4 - x^3
   

4. Subtract the result from the dividend. Subtract (x^4 - x^3) from (x^4 - 5x^3). This leaves -4x^3.

       x^3 ________
   x-1 | x^4 - 5x^3 + 10x^2 - 3x - 2 
         x^4 - x^3
         -------
             -4x^3
   

5. Bring down the next term of the dividend. Bring down the +10x^2 term.

       x^3 ________
   x-1 | x^4 - 5x^3 + 10x^2 - 3x - 2 
         x^4 - x^3
         -------
             -4x^3 + 10x^2
   

6. Repeat steps 2-5. Divide the new leading term (-4x^3) by the leading term of the divisor (x), which gives -4x^2. Write this quotient above the +10x^2 term. Multiply (x-1) by -4x^2, which gives -4x^3 + 4x^2. Write this result below the -4x^3 + 10x^2 term. Subtract the two, bringing down the next term (-3x).

       x^3 - 4x^2 _______
   x-1 | x^4 - 5x^3 + 10x^2 - 3x - 2 
         x^4 - x^3
         -------
             -4x^3 + 10x^2
             -4x^3 + 4x^2
             ---------
                     6x^2 - 3x
   

7. Continue repeating steps 2-5. Divide 6x^2 by x, which gives 6x. Multiply (x-1) by 6x, which gives 6x^2 - 6x. Subtract this from 6x^2 - 3x, bringing down the last term (-2).

       x^3 - 4x^2 + 6x _______
   x-1 | x^4 - 5x^3 + 10x^2 - 3x - 2 
         x^4 - x^3
         -------
             -4x^3 + 10x^2
             -4x^3 + 4x^2
             ---------
                     6x^2 - 3x
                     6x^2 - 6x
                     --------
                            3x - 2
   

8. Final step. Divide 3x by x, which gives 3. Multiply (x-1) by 3, which gives 3x - 3. Subtract this from 3x - 2, resulting in a remainder of 1.

       x^3 - 4x^2 + 6x + 3 _______
   x-1 | x^4 - 5x^3 + 10x^2 - 3x - 2 
         x^4 - x^3
         -------
             -4x^3 + 10x^2
             -4x^3 + 4x^2
             ---------
                     6x^2 - 3x
                     6x^2 - 6x
                     --------
                            3x - 2
                            3x - 3
                            -------
                                  1
   

Conclusion:

Therefore, the quotient of dividing (x^4 - 5x^3 + 10x^2 - 3x - 2) by (x-1) is x^3 - 4x^2 + 6x + 3 with a remainder of 1. We can express this result as:

(x^4 - 5x^3 + 10x^2 - 3x - 2) / (x-1) = x^3 - 4x^2 + 6x + 3 + 1/(x-1)