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(x^4-2x^3-29x^2-43x+8)/(x-7)

6 min read Jun 17, 2024
(x^4-2x^3-29x^2-43x+8)/(x-7)

Dividing Polynomials: A Step-by-Step Guide

This article will walk you through the process of dividing the polynomial (x^4 - 2x^3 - 29x^2 - 43x + 8) by (x - 7) using polynomial long division.

Step 1: Setting Up the Division

Begin by setting up the long division problem:

             ________
x - 7 | x^4 - 2x^3 - 29x^2 - 43x + 8

Step 2: Dividing the Leading Terms

  • Divide the leading term of the dividend (x^4) by the leading term of the divisor (x). This gives us x^3.
  • Write x^3 above the x^3 term in the dividend.
             x^3       
x - 7 | x^4 - 2x^3 - 29x^2 - 43x + 8

Step 3: Multiply and Subtract

  • Multiply the divisor (x - 7) by the term we just wrote above the line (x^3). This gives us x^4 - 7x^3.
  • Write this result below the dividend and subtract.
             x^3       
x - 7 | x^4 - 2x^3 - 29x^2 - 43x + 8 
       -(x^4 - 7x^3)
       ------------------
                 5x^3 - 29x^2

Step 4: Repeat the Process

  • Bring down the next term of the dividend (-29x^2).
  • Divide the leading term of the new dividend (5x^3) by the leading term of the divisor (x). This gives us 5x^2.
  • Write 5x^2 above the line.
             x^3 + 5x^2     
x - 7 | x^4 - 2x^3 - 29x^2 - 43x + 8 
       -(x^4 - 7x^3)
       ------------------
                 5x^3 - 29x^2
  • Multiply the divisor (x - 7) by 5x^2 to get 5x^3 - 35x^2.
  • Subtract this result from the current line.
             x^3 + 5x^2     
x - 7 | x^4 - 2x^3 - 29x^2 - 43x + 8 
       -(x^4 - 7x^3)
       ------------------
                 5x^3 - 29x^2
                 -(5x^3 - 35x^2)
                 ------------------
                         6x^2 - 43x

Step 5: Continue the Division

  • Bring down the next term of the dividend (-43x).
  • Divide the leading term of the new dividend (6x^2) by the leading term of the divisor (x). This gives us 6x.
  • Write 6x above the line.
             x^3 + 5x^2 + 6x    
x - 7 | x^4 - 2x^3 - 29x^2 - 43x + 8 
       -(x^4 - 7x^3)
       ------------------
                 5x^3 - 29x^2
                 -(5x^3 - 35x^2)
                 ------------------
                         6x^2 - 43x
  • Multiply the divisor (x - 7) by 6x to get 6x^2 - 42x.
  • Subtract this result from the current line.
             x^3 + 5x^2 + 6x    
x - 7 | x^4 - 2x^3 - 29x^2 - 43x + 8 
       -(x^4 - 7x^3)
       ------------------
                 5x^3 - 29x^2
                 -(5x^3 - 35x^2)
                 ------------------
                         6x^2 - 43x 
                         -(6x^2 - 42x)
                         ------------------
                                  -x + 8

Step 6: Final Steps

  • Bring down the last term of the dividend (8).
  • Divide the leading term of the new dividend (-x) by the leading term of the divisor (x). This gives us -1.
  • Write -1 above the line.
             x^3 + 5x^2 + 6x - 1   
x - 7 | x^4 - 2x^3 - 29x^2 - 43x + 8 
       -(x^4 - 7x^3)
       ------------------
                 5x^3 - 29x^2
                 -(5x^3 - 35x^2)
                 ------------------
                         6x^2 - 43x 
                         -(6x^2 - 42x)
                         ------------------
                                  -x + 8
  • Multiply the divisor (x - 7) by -1 to get -x + 7.
  • Subtract this result from the current line.
             x^3 + 5x^2 + 6x - 1   
x - 7 | x^4 - 2x^3 - 29x^2 - 43x + 8 
       -(x^4 - 7x^3)
       ------------------
                 5x^3 - 29x^2
                 -(5x^3 - 35x^2)
                 ------------------
                         6x^2 - 43x 
                         -(6x^2 - 42x)
                         ------------------
                                  -x + 8
                                  -(-x + 7)
                                  ------------
                                       1

Solution

Therefore, the result of dividing (x^4 - 2x^3 - 29x^2 - 43x + 8) by (x - 7) is:

(x^3 + 5x^2 + 6x - 1) with a remainder of 1.

This can also be written as:

(x^4 - 2x^3 - 29x^2 - 43x + 8) / (x - 7) = (x^3 + 5x^2 + 6x - 1) + 1/(x - 7)

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