Question

Express in simplest form by dividing x^2+8x-48/x^2-144/(x-4)

Answer 1

To simplify the expression \(\frac{x^2 + 8x - 48}{x^2 - 144} \div (x - 4)\), we can begin by rewriting the division as multiplication by the reciprocal:

\[
\frac{x^2 + 8x - 48}{x^2 - 144} \cdot \frac{1}{x - 4} = \frac{x^2 + 8x - 48}{x^2 - 144} \cdot \frac{1}{x - 4}
\]

Next, we'll factor each part of the expression.

1. **Factor the numerator \( x^2 + 8x - 48 \):**

To factor \( x^2 + 8x - 48 \), we look for two numbers that multiply to \(-48\) and add to \(8\). These numbers are \(12\) and \(-4\). Thus:

\[
x^2 + 8x - 48 = (x + 12)(x - 4)
\]

2. **Factor the denominator \( x^2 - 144 \):**

This is a difference of squares, which can be factored as:

\[
x^2 - 144 = (x - 12)(x + 12)
\]

Now, substituting these factored forms back into the expression, we have:

\[
\frac{(x + 12)(x - 4)}{(x - 12)(x + 12)} \cdot \frac{1}{x - 4}
\]

Next, we can simplify the expression. The \((x - 4)\) in the numerator and denominator cancels out (as long as \(x \neq 4\)):

\[
\frac{(x + 12)}{(x - 12)}
\]

Therefore, the expression simplifies to:

\[
\frac{x + 12}{x - 12} \quad \text{(for } x \neq 4)
\]

In its simplest form, the final answer is:

\[
\frac{x + 12}{x - 12} \quad \text{for } x \neq 4 \text{ and } x \neq -12
\] (since \(x + 12\) is zero when \(x = -12\), which would also make the original expression undefined).