(x^3+x^2+3x-4)/(x^2+2x+1) Long Division

Jasonbradley

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Jun 17, 2024

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Long Division of Polynomials: (x^3+x^2+3x-4)/(x^2+2x+1)

Long division of polynomials is a process used to divide a polynomial by another polynomial of a lower degree. Here's how to perform the long division of (x^3+x^2+3x-4) by (x^2+2x+1):

1. Set up the Division:

Write the problem in the standard long division format:

          _________
x^2+2x+1 | x^3 + x^2 + 3x - 4 

2. Divide the Leading Terms:

Divide the leading term of the dividend (x^3) by the leading term of the divisor (x^2). This gives you x. Write this above the division line, aligned with the x^3 term.

          x      
x^2+2x+1 | x^3 + x^2 + 3x - 4 

3. Multiply the Divisor:

Multiply the divisor (x^2+2x+1) by the term you just wrote (x). Write the result below the dividend.

          x      
x^2+2x+1 | x^3 + x^2 + 3x - 4 
          x^3 + 2x^2 + x

4. Subtract:

Subtract the line you just wrote from the dividend.

          x      
x^2+2x+1 | x^3 + x^2 + 3x - 4 
          x^3 + 2x^2 + x
          -------------
               -x^2 + 2x - 4 

5. Bring Down the Next Term:

Bring down the next term from the dividend (-4).

          x      
x^2+2x+1 | x^3 + x^2 + 3x - 4 
          x^3 + 2x^2 + x
          -------------
               -x^2 + 2x - 4 

6. Repeat Steps 2-5:

Now focus on the new dividend (-x^2 + 2x - 4).

• Divide the leading term of the new dividend (-x^2) by the leading term of the divisor (x^2). This gives you -1. Write this above the division line, aligned with the x^2 term.

          x - 1  
x^2+2x+1 | x^3 + x^2 + 3x - 4 
          x^3 + 2x^2 + x
          -------------
               -x^2 + 2x - 4 

• Multiply the divisor (x^2+2x+1) by -1. Write the result below the new dividend.

          x - 1  
x^2+2x+1 | x^3 + x^2 + 3x - 4 
          x^3 + 2x^2 + x
          -------------
               -x^2 + 2x - 4 
               -x^2 - 2x - 1

• Subtract.

          x - 1  
x^2+2x+1 | x^3 + x^2 + 3x - 4 
          x^3 + 2x^2 + x
          -------------
               -x^2 + 2x - 4 
               -x^2 - 2x - 1
               -------------
                      4x - 3

7. Stop when the Degree of the Remainder is Less than the Degree of the Divisor:

The degree of the remainder (4x-3) is 1, which is less than the degree of the divisor (x^2+2x+1). Therefore, we stop here.

Result:

The result of the long division is:

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

This means the original polynomial can be expressed as:

x^3 + x^2 + 3x - 4 = (x^2 + 2x + 1)(x - 1) + (4x - 3)