(2x^3+4x^2-3x-6)/(x+3)

Jasonbradley

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Jun 16, 2024

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Dividing Polynomials: (2x^3 + 4x^2 - 3x - 6) / (x + 3)

This article will guide you through the process of dividing the polynomial 2x^3 + 4x^2 - 3x - 6 by x + 3 using polynomial long division.

Polynomial Long Division

Polynomial long division is similar to the long division you learned in elementary school, but instead of numbers, we are dealing with polynomials. Here's how to perform the division:

1. Set up the division: Write the dividend (2x^3 + 4x^2 - 3x - 6) inside the division symbol and the divisor (x + 3) outside.

       ____________
   x + 3 | 2x^3 + 4x^2 - 3x - 6
   

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

       2x^2        
   x + 3 | 2x^3 + 4x^2 - 3x - 6
   

3. Multiply the divisor: Multiply the term we just wrote (2x^2) by the entire divisor (x + 3). This gives us 2x^3 + 6x^2. Write this result below the dividend.

       2x^2        
   x + 3 | 2x^3 + 4x^2 - 3x - 6
          2x^3 + 6x^2 
   

4. Subtract: Subtract the result (2x^3 + 6x^2) from the corresponding terms of the dividend. This leaves us with -2x^2 - 3x.

       2x^2        
   x + 3 | 2x^3 + 4x^2 - 3x - 6
          2x^3 + 6x^2 
          ------------
              -2x^2 - 3x
   

5. Bring down the next term: Bring down the next term of the dividend (-3x). This gives us -2x^2 - 3x - 6.

       2x^2        
   x + 3 | 2x^3 + 4x^2 - 3x - 6
          2x^3 + 6x^2 
          ------------
              -2x^2 - 3x - 6
   

6. Repeat steps 2-5: Now, repeat the process from step 2. Divide the leading term of the new dividend (-2x^2) by the leading term of the divisor (x), which gives us -2x. Write this above the division symbol. Multiply -2x by the divisor (x + 3) and subtract the result from the current dividend.

       2x^2 - 2x     
   x + 3 | 2x^3 + 4x^2 - 3x - 6
          2x^3 + 6x^2 
          ------------
              -2x^2 - 3x - 6
              -2x^2 - 6x
              -----------
                      3x - 6 
   

7. Repeat again: Bring down the last term of the dividend (-6). Divide the new leading term (3x) by the leading term of the divisor (x) to get 3. Write this above the division symbol. Multiply 3 by the divisor (x + 3) and subtract the result from the current dividend.

       2x^2 - 2x + 3 
   x + 3 | 2x^3 + 4x^2 - 3x - 6
          2x^3 + 6x^2 
          ------------
              -2x^2 - 3x - 6
              -2x^2 - 6x
              -----------
                      3x - 6
                      3x + 9
                      -------
                          -15
   

8. The remainder: The result of the division is 2x^2 - 2x + 3 with a remainder of -15. We can write this as:

   (2x^3 + 4x^2 - 3x - 6) / (x + 3) = 2x^2 - 2x + 3 - 15/(x + 3)

Conclusion

By using polynomial long division, we have successfully divided (2x^3 + 4x^2 - 3x - 6) by (x + 3). The result is a quotient of 2x^2 - 2x + 3 and a remainder of -15. This process is essential for simplifying polynomials and solving problems involving polynomial equations.