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

Jasonbradley

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Jun 17, 2024

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

Long division is a fundamental technique in algebra for dividing polynomials. Let's illustrate this process with the example of dividing (x^3 + 3x^2 + 3x + 2) by (x - 1).

Steps for Long Division

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

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

2. Divide the leading terms: Divide the leading term of the dividend (x^3) by the leading term of the divisor (x). This gives us x^2. Write this term above the division symbol, aligning it with the x^2 term in the dividend.

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

3. Multiply and subtract: Multiply the quotient term (x^2) by the divisor (x - 1), which gives us x^3 - x^2. Write this result below the dividend, aligning the terms. Subtract this result from the dividend.

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

4. Bring down the next term: Bring down the next term from the dividend (3x).

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

5. Repeat the process: Repeat steps 2-4 with the new dividend (4x^2 + 3x). Divide the leading term (4x^2) by the leading term of the divisor (x), resulting in 4x. Multiply 4x by the divisor (x - 1), which gives us 4x^2 - 4x. Subtract this from the current dividend.

       x^2 + 4x   
   x - 1 | x^3 + 3x^2 + 3x + 2
           x^3 - x^2
           -------
               4x^2 + 3x 
               4x^2 - 4x
               -------
                       7x + 2
   

6. Bring down the next term: Bring down the next term from the dividend (2).

       x^2 + 4x   
   x - 1 | x^3 + 3x^2 + 3x + 2
           x^3 - x^2
           -------
               4x^2 + 3x 
               4x^2 - 4x
               -------
                       7x + 2 
   

7. Repeat the process: Divide the leading term of the current dividend (7x) by the leading term of the divisor (x), resulting in 7. Multiply 7 by the divisor (x - 1), which gives us 7x - 7. Subtract this from the current dividend.

       x^2 + 4x + 7  
   x - 1 | x^3 + 3x^2 + 3x + 2
           x^3 - x^2
           -------
               4x^2 + 3x 
               4x^2 - 4x
               -------
                       7x + 2 
                       7x - 7
                       -------
                           9 
   

8. The remainder: The final result is the quotient (x^2 + 4x + 7) and a remainder of 9.

Result

Therefore, (x^3 + 3x^2 + 3x + 2) / (x - 1) = x^2 + 4x + 7 + 9/(x - 1).