Questions: Subtract and simplify. w/(w^2-2w-35) - 8/(w+5)

Solution Steps

We start with the expression \[ \frac{w}{w^{2}-2w-35}-\frac{8}{w+5}. \] First, we factor the denominator of the first fraction, \( w^{2}-2w-35 \). The factorization gives us: \[ w^{2}-2w-35 = (w - 7)(w + 5). \]

Using the factored form of the denominator, we can rewrite the first expression as: \[ \frac{w}{(w - 7)(w + 5)}. \]

To perform the subtraction, we need a common denominator. The common denominator for both fractions is: \[ (w - 7)(w + 5)^{2}. \]

We rewrite both fractions with the common denominator: \[ \frac{w}{(w - 7)(w + 5)} \cdot \frac{(w + 5)}{(w + 5)} - \frac{8}{(w + 5)} \cdot \frac{(w - 7)}{(w - 7)}. \] This results in: \[ \frac{w(w + 5)}{(w - 7)(w + 5)^{2}} - \frac{8(w - 7)}{(w - 7)(w + 5)^{2}}. \]

Now we can subtract the two fractions: \[ \frac{w(w + 5) - 8(w - 7)}{(w - 7)(w + 5)^{2}}. \]

Finally, we simplify the numerator: \[ w(w + 5) - 8(w - 7) = w^{2} + 5w - 8w + 56 = w^{2} - 3w + 56. \] Thus, the simplified result is: \[ \frac{7(8 - w)}{w^{2} - 2w - 35}. \]

Final Answer