Express in simplest form: x2−6x−16/x2−64÷2x+4/x2+x−56

(1 point)
Responses

−x+72
the fraction with numerator negative x plus 7 and denominator 2

x−72
the fraction with numerator x minus 7 and denominator 2

x+72
the fraction with numerator x plus 7 and denominator 2

(x−7)(x+8)2(x−8)

1 answer

To simplify the expression \( \frac{x^2 - 6x - 16}{x^2 - 64} \div \frac{2x + 4}{x^2 + x - 56} \), we can start by rewriting the division as multiplication by the reciprocal.

This gives us:

\[ \frac{x^2 - 6x - 16}{x^2 - 64} \times \frac{x^2 + x - 56}{2x + 4} \]

Next, let's factor each component of the expression.

  1. Factor \( x^2 - 6x - 16 \): \[ x^2 - 6x - 16 = (x - 8)(x + 2) \]

  2. Factor \( x^2 - 64 \) (difference of squares): \[ x^2 - 64 = (x - 8)(x + 8) \]

  3. Factor \( 2x + 4 \): \[ 2x + 4 = 2(x + 2) \]

  4. Factor \( x^2 + x - 56 \): \[ x^2 + x - 56 = (x - 7)(x + 8) \]

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

\[ \frac{(x - 8)(x + 2)}{(x - 8)(x + 8)} \times \frac{(x - 7)(x + 8)}{2(x + 2)} \]

Now, we can simplify step-by-step:

  • Cancel \( (x - 8) \) from the numerator and the denominator.
  • Cancel \( (x + 2) \) from the numerator and the denominator.

This leaves us with:

\[ \frac{(x - 7)(x + 8)}{2} \]

Now, this can be rewritten as:

\[ \frac{(x - 7)(x + 8)}{2} \]

Thus, the simplest form of the original expression is:

\[ \frac{(x - 7)(x + 8)}{2} \]

Since this matches the second response option given (the fraction with numerator \( x - 7 \) and denominator \( 2 \)), the correct answer is:

the fraction with numerator \( x - 7 \) and denominator \( 2 \).