(x^2+7x+12)/(x+2)

Jasonbradley

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Jun 17, 2024

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Simplifying the Expression: (x^2 + 7x + 12) / (x + 2)

This expression represents a rational function, where the numerator is a quadratic polynomial and the denominator is a linear polynomial. To simplify this expression, we can use the technique of polynomial long division.

Steps for Polynomial Long Division:

1. Set up the division:

        _______
   x + 2 | x^2 + 7x + 12 
   

2. Divide the leading terms:

   • The leading term of the divisor (x + 2) is 'x'.
   • The leading term of the dividend (x^2 + 7x + 12) is 'x^2'.
   • x^2 / x = x. Write 'x' above the line.

        x______
   x + 2 | x^2 + 7x + 12 
   

3. Multiply the quotient by the divisor:

   • x * (x + 2) = x^2 + 2x

4. Subtract the result from the dividend:

        x______
   x + 2 | x^2 + 7x + 12 
           -(x^2 + 2x)
           -------
                 5x + 12
   

5. Bring down the next term:

        x______
   x + 2 | x^2 + 7x + 12 
           -(x^2 + 2x)
           -------
                 5x + 12
   

6. Repeat steps 2-5:

   • The leading term of the new dividend (5x + 12) is '5x'.
   • 5x / x = 5. Write '+ 5' above the line.

        x + 5___
   x + 2 | x^2 + 7x + 12 
           -(x^2 + 2x)
           -------
                 5x + 12
                 -(5x + 10)
                 -------
                       2
   

7. The remainder is 2.

Result:

The simplified expression is:

(x^2 + 7x + 12) / (x + 2) = x + 5 + 2/(x + 2)

This means the original expression can be rewritten as a linear expression (x + 5) plus a rational term (2/(x + 2)).

Note: This simplification is valid for all values of 'x' except for x = -2, as this would make the denominator zero, leading to an undefined expression.