(2x^3+5x^2+9)/(x+3)

Jasonbradley

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Jun 16, 2024

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Dividing Polynomials: (2x^3 + 5x^2 + 9) / (x + 3)

This article will walk through the process of dividing the polynomial 2x^3 + 5x^2 + 9 by the binomial x + 3. We will utilize the long division method to achieve this.

Long Division Steps

1. Set up the division problem:

         __________
   x + 3 | 2x^3 + 5x^2 + 0x + 9 
   

   • Notice that we've added a placeholder term 0x for the x term, as it was not explicitly present in the original polynomial.

2. Divide the leading terms:

   • (2x^3) / (x) = 2x^2

   • Write this result above the x^2 term in the quotient:

         2x^2     
         __________
   x + 3 | 2x^3 + 5x^2 + 0x + 9 
   

3. Multiply the divisor by the result:

   • (2x^2) * (x + 3) = 2x^3 + 6x^2

   • Write this result below the dividend:

         2x^2     
         __________
   x + 3 | 2x^3 + 5x^2 + 0x + 9 
          2x^3 + 6x^2
   

4. Subtract:

   • Subtract the terms vertically:

         2x^2     
         __________
   x + 3 | 2x^3 + 5x^2 + 0x + 9 
          2x^3 + 6x^2
          ---------
                -x^2  + 0x 
   

5. Bring down the next term:

   • Bring down the 0x term from the dividend:

         2x^2     
         __________
   x + 3 | 2x^3 + 5x^2 + 0x + 9 
          2x^3 + 6x^2
          ---------
                -x^2  + 0x + 9 
   

6. Repeat steps 2-5:

   • Divide the leading term of the new dividend (-x^2) by the leading term of the divisor (x):

     • (-x^2) / (x) = -x

   • Write the result above the x term in the quotient:

         2x^2 - x      
         __________
     

   x + 3 | 2x^3 + 5x^2 + 0x + 9 2x^3 + 6x^2 --------- -x^2 + 0x + 9 -x^2 - 3x

   
   * Multiply the divisor by the new result:
   
   * **(-x) * (x + 3) = -x^2 - 3x**
   
   * Subtract:
   
   

      2x^2 - x      
      __________
   

   x + 3 | 2x^3 + 5x^2 + 0x + 9 2x^3 + 6x^2 --------- -x^2 + 0x + 9 -x^2 - 3x --------- 3x + 9

   
   * Bring down the last term:
   
   

      2x^2 - x      
      __________
   

   x + 3 | 2x^3 + 5x^2 + 0x + 9 2x^3 + 6x^2 --------- -x^2 + 0x + 9 -x^2 - 3x --------- 3x + 9

7. Final step:

   • Divide the new leading term (3x) by the leading term of the divisor (x):

     • (3x) / (x) = 3

   • Write the result above the constant term in the quotient:

         2x^2 - x + 3     
         __________
     

   x + 3 | 2x^3 + 5x^2 + 0x + 9 2x^3 + 6x^2 --------- -x^2 + 0x + 9 -x^2 - 3x --------- 3x + 9 3x + 9

   
   * Multiply the divisor by the result:
   
   * **(3) * (x + 3) = 3x + 9**
   
   * Subtract:
   
   

      2x^2 - x + 3     
      __________
   

   x + 3 | 2x^3 + 5x^2 + 0x + 9 2x^3 + 6x^2 --------- -x^2 + 0x + 9 -x^2 - 3x --------- 3x + 9 3x + 9 --------- 0

Conclusion

Therefore, the result of dividing (2x^3 + 5x^2 + 9) by (x + 3) is 2x^2 - x + 3. This means the original polynomial can be rewritten as: (x + 3)(2x^2 - x + 3).