(a^2-8a-26)/(a+2)

Jasonbradley

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Jun 16, 2024

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Simplifying the Expression (a^2 - 8a - 26) / (a + 2)

This expression represents a rational expression, where the numerator is a quadratic polynomial and the denominator is a linear polynomial. We can simplify this expression by performing polynomial long division.

1. Setting up the Long Division

We set up the long division as follows:

          a - 10
      a + 2 | a^2 - 8a - 26
             -(a^2 + 2a)
              -10a - 26
              -(-10a - 20)
                     -6 

2. Performing the Division

• Step 1: Divide the leading term of the dividend (a^2) by the leading term of the divisor (a). This gives us 'a'. Write 'a' above the line as the first term of the quotient.
• Step 2: Multiply the divisor (a + 2) by the term 'a' we just found, and write the result (a^2 + 2a) below the dividend.
• Step 3: Subtract the result from the dividend. This leaves us with -10a - 26.
• Step 4: Bring down the next term (-26) to form the new dividend.
• Step 5: Divide the leading term of the new dividend (-10a) by the leading term of the divisor (a). This gives us '-10'. Write '-10' above the line as the next term of the quotient.
• Step 6: Multiply the divisor (a + 2) by the term '-10', and write the result (-10a - 20) below the new dividend.
• Step 7: Subtract the result from the new dividend. This leaves us with -6.

3. The Result

The final result of the long division is:

(a^2 - 8a - 26) / (a + 2) = a - 10 - 6/(a + 2)

Important Note: The remainder of the division is -6, which is written over the original divisor (a + 2). This indicates that the original expression cannot be completely simplified. However, we have successfully expressed the original expression as the sum of a polynomial (a - 10) and a rational expression (-6/(a + 2)).