Questions: Divide and simplify the following rational expression. (x^2-15x+56)/(x^2+17x+72) ÷ (x-8)/(x+9) Give your answer as a reduced rational expression.

Solution Steps

We start by factoring the polynomials in the given rational expression:

1. The numerator \( x^2 - 15x + 56 \) factors to: \[ (x - 8)(x - 7) \]

2. The denominator \( x^2 + 17x + 72 \) factors to: \[ (x + 8)(x + 9) \]

3. The expression \( x - 8 \) remains as is since it is already factored.

4. The expression \( x + 9 \) also remains as is.

The original expression can be rewritten using the factored forms: \[ \frac{(x - 8)(x - 7)}{(x + 8)(x + 9)} \div \frac{x - 8}{x + 9} \] This division can be transformed into multiplication by taking the reciprocal: \[ \frac{(x - 8)(x - 7)}{(x + 8)(x + 9)} \cdot \frac{x + 9}{x - 8} \]

Now, we can simplify the expression:

• The \( (x - 8) \) terms in the numerator and denominator cancel out: \[ \frac{(x - 7)(x + 9)}{(x + 8)} \]

The simplified expression is: \[ \frac{x - 7}{x + 8} \]

Final Answer