(3x^3+9x^2+8x+4)/(x+2)

Jasonbradley

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Jun 16, 2024

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Dividing Polynomials: A Step-by-Step Guide

This article will guide you through the process of dividing the polynomial (3x^3 + 9x^2 + 8x + 4) by (x + 2).

Long Division Method

We will use the long division method to perform this division. Here's a breakdown of the steps:

1. Set up the problem: Write the division problem in the traditional long division format:

        ___________
   x + 2 | 3x^3 + 9x^2 + 8x + 4
   

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

        3x^2 ______
   x + 2 | 3x^3 + 9x^2 + 8x + 4
   

3. Multiply the quotient term by the divisor: Multiply 3x^2 by (x + 2) to get 3x^3 + 6x^2. Write this result below the dividend.

        3x^2 ______
   x + 2 | 3x^3 + 9x^2 + 8x + 4
           3x^3 + 6x^2 
   

4. Subtract: Subtract the result from step 3 from the dividend. Be careful to distribute the minus sign.

        3x^2 ______
   x + 2 | 3x^3 + 9x^2 + 8x + 4
           3x^3 + 6x^2 
           ---------
                 3x^2 + 8x
   

5. Bring down the next term: Bring down the next term (8x) from the dividend.

        3x^2 ______
   x + 2 | 3x^3 + 9x^2 + 8x + 4
           3x^3 + 6x^2 
           ---------
                 3x^2 + 8x + 4
   

6. Repeat steps 2-5: Repeat the process: Divide the new leading term (3x^2) by the divisor's leading term (x) to get 3x. Write this above the dividend, aligning it with the x term.

        3x^2 + 3x ______
   x + 2 | 3x^3 + 9x^2 + 8x + 4
           3x^3 + 6x^2 
           ---------
                 3x^2 + 8x + 4
                 3x^2 + 6x
   

   Multiply 3x by (x + 2) to get 3x^2 + 6x. Subtract this from the previous result:

        3x^2 + 3x ______
   x + 2 | 3x^3 + 9x^2 + 8x + 4
           3x^3 + 6x^2 
           ---------
                 3x^2 + 8x + 4
                 3x^2 + 6x 
                 -------
                       2x + 4
   

7. Final step: Bring down the last term (4). Repeat the process: Divide 2x by x to get 2. Write this above the dividend, aligning it with the constant term.

        3x^2 + 3x + 2 ______
   x + 2 | 3x^3 + 9x^2 + 8x + 4
           3x^3 + 6x^2 
           ---------
                 3x^2 + 8x + 4
                 3x^2 + 6x 
                 -------
                       2x + 4
                       2x + 4
   

   Multiply 2 by (x + 2) to get 2x + 4. Subtract this from the previous result:

        3x^2 + 3x + 2 ______
   x + 2 | 3x^3 + 9x^2 + 8x + 4
           3x^3 + 6x^2 
           ---------
                 3x^2 + 8x + 4
                 3x^2 + 6x 
                 -------
                       2x + 4
                       2x + 4
                       -------
                            0
   

8. Result: Since the remainder is 0, we have successfully divided the polynomial. The quotient is 3x^2 + 3x + 2.

Conclusion:

We have demonstrated how to divide the polynomial (3x^3 + 9x^2 + 8x + 4) by (x + 2) using the long division method. The quotient of this division is 3x^2 + 3x + 2.