Express in simplest form: x2+8x−48x2−144 ÷(4−x)(1 point)

Question

Responses

−(x+4)(x−4)(x+12)
the fraction with numerator negative open paren x plus 4 close paren and denominator open paren x minus 4 close paren times open paren x plus 12 close paren

−1x−12
the fraction with numerator negative 1 and denominator x minus 12

1x−12
the fraction with numerator 1 and denominator x minus 12

(x−4)(x+4)x−12
the fraction with numerator open paren x minus 4 close paren times open paren x plus 4 close paren and denominator x minus 12

Answer 1

To simplify the expression \( \frac{x^2 + 8x - 48}{x^2 - 144} \div (4 - x) \), we start by rewriting the expression in a multiplication form:

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

Next, we simplify each part in the expression.

1. Factor the numerator and denominator of the first fraction:

• The numerator, \( x^2 + 8x - 48 \), factors as: \[ x^2 + 8x - 48 = (x + 12)(x - 4) \]

• The denominator, \( x^2 - 144 \), is a difference of squares, which factors as: \[ x^2 - 144 = (x - 12)(x + 12) \]

Now substituting these factors into our expression, we have: \[ \frac{(x + 12)(x - 4)}{(x - 12)(x + 12)} \cdot \frac{1}{4 - x} \]

2. Simplify the fraction:

The \( (x + 12) \) cancels out from the numerator and the denominator: \[ \frac{x - 4}{(x - 12)} \cdot \frac{1}{4 - x} \]

Notice that \( 4 - x \) can be expressed as \( -(x - 4) \). This means we can rewrite this part: \[ \frac{x - 4}{(x - 12)} \cdot \frac{1}{-(x - 4)} = \frac{1}{-(x - 12)} = \frac{-1}{x - 12} \]

Thus, the entire expression simplifies to: \[ -\frac{1}{x - 12} \]

Therefore, the final answer in simplest form is:

\[ -\frac{1}{x - 12} \]

So the correct response from the options provided is: −1/(x−12)