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

Jasonbradley

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Jun 16, 2024

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Long Division of Polynomials: (3x^3+9x^2+8x+4)/(x+2)

Long division of polynomials is a method used to divide a polynomial by another polynomial of a lower or equal degree. This process is similar to long division of numbers, but with variables and exponents.

Here's how to perform the long division of (3x^3+9x^2+8x+4) by (x+2):

1. Set up the division:

           ________
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): 3x^3 / x = 3x^2
• Write the result (3x^2) above the dividend.

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

3. Multiply the divisor by the quotient term:

• Multiply the divisor (x+2) by the quotient term (3x^2): (x+2) * 3x^2 = 3x^3 + 6x^2
• Write the result below the dividend.

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

4. Subtract:

• Subtract the product (3x^3 + 6x^2) from the dividend (3x^3 + 9x^2 + 8x + 4): (3x^3 + 9x^2 + 8x + 4) - (3x^3 + 6x^2) = 3x^2 + 8x + 4

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

5. Bring down the next term:

• Bring down the next term of the dividend (8x).

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

6. Repeat steps 2-5:

• Divide the new leading term (3x^2) by the divisor's leading term (x): 3x^2 / x = 3x
• Multiply the divisor (x+2) by the new quotient term (3x): (x+2) * 3x = 3x^2 + 6x
• Subtract the product (3x^2 + 6x) from the previous result (3x^2 + 8x + 4): (3x^2 + 8x + 4) - (3x^2 + 6x) = 2x + 4
• Bring down the next term (4): 2x + 4 + 4 = 2x + 8

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

7. Final subtraction:

• Subtract the final product (2x + 4) from the previous result (2x + 8): (2x + 8) - (2x + 4) = 4

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

Result:

The quotient is 3x^2 + 3x and the remainder is 4.

Therefore, (3x^3+9x^2+8x+4)/(x+2) = 3x^2 + 3x + 4/(x+2).