Question: Simplify: \((8 - x) \div \frac{x^2 - 64}{4} \rightarrow (x - 8)\) Options: 1. \(\frac{-8}{x…

Simplify: \((8 - x) \div \frac{x^2 - 64}{4} \rightarrow (x - 8)\)

Options:

1. \(\frac{-8}{x + 4}\)
2. \(\frac{-8}{x + 8}\)
3. \(\frac{-4}{x - 8}\)
4. \(4(x + 8)\)
5. \(\frac{-4}{x + 8}\)

Solution

To simplify \( (8 - x) \div \frac{x^2 - 64}{4} \): First, recognize that dividing by a fraction is the same as multiplying by its reciprocal: \[ (8 - x) \times \frac{4}{x^2 - 64} \] Factor \( x^2 - 64 \) as a difference of squares: \[ x^2 - 64 = (x - 8)(x + 8) \] Substitute the factorization: \[ (8 - x) \times \frac{4}{(x - 8)(x + 8)} \] Notice that \( 8 - x = -(x - 8) \): \[ -(x - 8) \times \frac{4}{(x - 8)(x + 8)} \] Cancel \( x - 8 \) in the numerator and denominator: \[ -\frac{4}{x + 8} \] The simplified expression is: \[ -\frac{4}{x + 8} \]