(4x^4-2x^3+x^2-5x+8)/(x^2-2x-1)

Jasonbradley

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Jun 16, 2024

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Polynomial Long Division: (4x^4-2x^3+x^2-5x+8)/(x^2-2x-1)

This article will walk through the process of performing polynomial long division on the expression (4x^4-2x^3+x^2-5x+8)/(x^2-2x-1).

Understanding Polynomial Long Division

Polynomial long division is a method for dividing polynomials, similar to the long division you may have learned for numbers. The goal is to find a quotient and a remainder.

Steps for Polynomial Long Division

1. Set up the division: Write the dividend (4x^4-2x^3+x^2-5x+8) inside the division symbol and the divisor (x^2-2x-1) outside.

        ____________
   x^2-2x-1 | 4x^4 - 2x^3 + x^2 - 5x + 8 
   

2. Divide the leading terms: Divide the leading term of the dividend (4x^4) by the leading term of the divisor (x^2). This gives us 4x^2, which we write above the division symbol.

        4x^2       
   x^2-2x-1 | 4x^4 - 2x^3 + x^2 - 5x + 8 
   

3. Multiply the divisor by the quotient term: Multiply (x^2-2x-1) by 4x^2, which gives us 4x^4 - 8x^3 - 4x^2. Write this result below the dividend.

        4x^2       
   x^2-2x-1 | 4x^4 - 2x^3 + x^2 - 5x + 8 
             4x^4 - 8x^3 - 4x^2 
   

4. Subtract: Subtract the result from the dividend. This gives us 6x^3 + 5x^2 - 5x.

        4x^2       
   x^2-2x-1 | 4x^4 - 2x^3 + x^2 - 5x + 8 
             4x^4 - 8x^3 - 4x^2 
             ------------------
                  6x^3 + 5x^2 - 5x
   

5. Bring down the next term: Bring down the next term from the dividend (-5x).

        4x^2       
   x^2-2x-1 | 4x^4 - 2x^3 + x^2 - 5x + 8 
             4x^4 - 8x^3 - 4x^2 
             ------------------
                  6x^3 + 5x^2 - 5x + 8
   

6. Repeat steps 2-5: Repeat the process, now dividing the leading term of the new dividend (6x^3) by the leading term of the divisor (x^2). This gives us 6x.

        4x^2 + 6x    
   x^2-2x-1 | 4x^4 - 2x^3 + x^2 - 5x + 8 
             4x^4 - 8x^3 - 4x^2 
             ------------------
                  6x^3 + 5x^2 - 5x + 8
                  6x^3 - 12x^2 - 6x 
   

   Continue multiplying and subtracting. Remember to bring down the next term from the dividend.

        4x^2 + 6x     
   x^2-2x-1 | 4x^4 - 2x^3 + x^2 - 5x + 8 
             4x^4 - 8x^3 - 4x^2 
             ------------------
                  6x^3 + 5x^2 - 5x + 8
                  6x^3 - 12x^2 - 6x 
                  ------------------
                         17x^2 + x + 8
   

7. Repeat the process again: Divide the leading term of the new dividend (17x^2) by the leading term of the divisor (x^2). This gives us 17.

        4x^2 + 6x + 17
   x^2-2x-1 | 4x^4 - 2x^3 + x^2 - 5x + 8 
             4x^4 - 8x^3 - 4x^2 
             ------------------
                  6x^3 + 5x^2 - 5x + 8
                  6x^3 - 12x^2 - 6x 
                  ------------------
                         17x^2 + x + 8
                         17x^2 - 34x - 17
   

8. Subtract and bring down: Subtract the result and bring down the next term (8). This leaves us with a final remainder of 35x + 25.

        4x^2 + 6x + 17
   x^2-2x-1 | 4x^4 - 2x^3 + x^2 - 5x + 8 
             4x^4 - 8x^3 - 4x^2 
             ------------------
                  6x^3 + 5x^2 - 5x + 8
                  6x^3 - 12x^2 - 6x 
                  ------------------
                         17x^2 + x + 8
                         17x^2 - 34x - 17
                         -----------------
                                35x + 25
   

Final Result

The quotient of the division is 4x^2 + 6x + 17, and the remainder is 35x + 25.

Therefore, we can write the original expression as:

(4x^4 - 2x^3 + x^2 - 5x + 8) / (x^2 - 2x - 1) = 4x^2 + 6x + 17 + (35x + 25) / (x^2 - 2x - 1)