Question

add or subtract, as indicated, and simplify the result.

\\\frac{p - 6}{p^2 + 18p + 81} + \frac{1}{p + 9} - \frac{2p + 3}{2p^2 + 15p - 27}\\

\\\frac{p - 6}{p^2 + 18p + 81} + \frac{1}{p + 9} - \frac{2p + 3}{2p^2 + 15p - 27} = \square\\
(simplify your answer.)

Explanation:

Factor the denominators

We factor each quadratic denominator to find the least common denominator.
Using the Rational Expressions knowledge point:

$$ p^2 + 18p + 81 = (p + 9)^2 $$

For \(2p^2 + 15p - 27\), we find factors of \(2 \times (-27) = -54\) that sum to \(15\), which are \(18\) and \(-3\):

$$ 2p^2 + 18p - 3p - 27 = 2p(p + 9) - 3(p + 9) = (2p - 3)(p + 9) $$

Find the least common denominator

We identify the least common denominator (LCD) from the factored forms.
Using the Rational Expressions knowledge point:
The factored denominators are \((p+9)^2\), \((p+9)\), and \((2p-3)(p+9)\).
The LCD is:

$$ \text{LCD} = (2p - 3)(p + 9)^2 $$

Rewrite fractions with the LCD

We multiply the numerator and denominator of each term by the missing factors.
Using the Rational Expressions knowledge point:

$$ \frac{p-6}{(p+9)^2} = \frac{(p-6)(2p-3)}{(2p-3)(p+9)^2} $$

$$ \frac{1}{p+9} = \frac{(2p-3)(p+9)}{(2p-3)(p+9)^2} $$

$$ \frac{2p+3}{(2p-3)(p+9)} = \frac{(2p+3)(p+9)}{(2p-3)(p+9)^2} $$

Combine and simplify the numerators

We expand and combine the numerators over the common denominator.
Using the Rational Expressions knowledge point:
Expand each term:

$$ (p-6)(2p-3) = 2p^2 - 15p + 18 $$

$$ (2p-3)(p+9) = 2p^2 + 15p - 27 $$

$$ (2p+3)(p+9) = 2p^2 + 21p + 27 $$

Combine them:

$$ (2p^2 - 15p + 18) + (2p^2 + 15p - 27) - (2p^2 + 21p + 27) $$

$$ = 2p^2 - 21p - 36 $$

Factor and simplify the final expression

We factor the combined numerator to see if any terms cancel.
Using the Rational Expressions knowledge point:
Factor \(2p^2 - 21p - 36\). We look for factors of \(2 \times (-36) = -72\) that sum to \(-21\), which are \(-24\) and \(3\):

$$ 2p^2 - 24p + 3p - 36 = 2p(p - 12) + 3(p - 12) = (2p + 3)(p - 12) $$

The simplified rational expression is:

$$ \frac{(2p + 3)(p - 12)}{(2p - 3)(p + 9)^2} $$

Answer:

Add or subtract, as indicated, and simplify the result.

\(\frac{p - 6}{p^2 + 18p + 81} + \frac{1}{p + 9} - \frac{2p + 3}{2p^2 + 15p - 27} =\) <blank>\(\frac{(2p + 3)(p - 12)}{(2p - 3)(p + 9)^2}\)</blank>
(Simplify your answer.)