(x^2+8x+15)/(x+5) Long Division

Jasonbradley

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Jun 17, 2024

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Long Division of Polynomials: (x^2 + 8x + 15) / (x + 5)

Long division of polynomials is a method for dividing two polynomials. It is similar to the long division of numbers, but we work with variables and coefficients.

In this example, we will divide the polynomial (x^2 + 8x + 15) by the polynomial (x + 5). Here's how to do it:

1. Set up the division:

          _______
x + 5 | x^2 + 8x + 15 

2. Divide the leading terms:

• The leading term of the divisor (x + 5) is 'x'.
• The leading term of the dividend (x^2 + 8x + 15) is 'x^2'.
• Divide x^2 by x, which gives us 'x'.
• Write 'x' above the x^2 term in the quotient.

          x _______
x + 5 | x^2 + 8x + 15 

3. Multiply the quotient by the divisor:

• Multiply 'x' (the quotient term) by (x + 5), which gives us 'x^2 + 5x'.

          x _______
x + 5 | x^2 + 8x + 15 
         x^2 + 5x

4. Subtract:

• Subtract 'x^2 + 5x' from the dividend.

          x _______
x + 5 | x^2 + 8x + 15 
         x^2 + 5x
         -------
              3x 

5. Bring down the next term:

• Bring down the next term from the dividend, which is '+ 15'.

          x _______
x + 5 | x^2 + 8x + 15 
         x^2 + 5x
         -------
              3x + 15

6. Repeat steps 2-5:

• Divide the leading term of the new dividend ('3x') by the leading term of the divisor ('x'), which gives us '3'.
• Write '3' next to 'x' in the quotient.
• Multiply '3' by (x + 5), which gives us '3x + 15'.
• Subtract '3x + 15' from the dividend.

          x + 3
x + 5 | x^2 + 8x + 15 
         x^2 + 5x
         -------
              3x + 15
              3x + 15
              -------
                    0

7. Result:

Since the remainder is 0, the division is complete. Therefore:

(x^2 + 8x + 15) / (x + 5) = x + 3

Important Note: You can check your answer by multiplying the quotient (x + 3) by the divisor (x + 5). The result should be the original dividend (x^2 + 8x + 15).