Simplify the following rational expression.
$\frac{8 x-56}{x^2-2 x-35}$
To simplify the rational expression \(\frac{8x - 56}{x^2 - 2x - 35}\), we begin by examining both the numerator and the denominator separately to identify any common factors. A rational expression is simplified by factoring both parts and canceling common terms. We must also consider the domain, as the expression is undefined where the denominator equals zero.
First, we factor the numerator \(8x - 56\). We can see that both terms are divisible by 8:
\[8x - 56 = 8(x - 7).\]
Next, we factor the denominator \(x^2 - 2x - 35\). We look for two integers that multiply to -35 and add up to -2. Those integers are -7 and 5. Thus, the quadratic factors as:
\[x^2 - 2x - 35 = (x - 7)(x + 5).\]
Substituting these factored forms back into the original expression, we get:
\[\frac{8(x - 7)}{(x - 7)(x + 5)}.\]
Now we cancel the common factor \((x - 7)\) from both the numerator and the denominator. Note that this cancellation is valid as long as the common factor is not zero, which means \(x
eq 7\). Additionally, the original expression is undefined where the denominator is zero, meaning \(x^2 - 2x - 35 = 0\), which occurs when \(x = 7\) or \(x = -5\). Therefore, the simplification is valid for all \(x\) except \(x = 7\) and \(x = -5\).
After canceling, we are left with:
\[\frac{8}{x + 5}.\]
The simplified form of the rational expression \(\frac{8x - 56}{x^2 - 2x - 35}\) is \(\frac{8}{x + 5}\), provided that \(x
eq 7\) and \(x
eq -5\). This completes the simplification process.
The final answer is \(\boxed{\frac{8}{x + 5}}\).
Simplifying rational expressions is a vital skill in algebra used to model real-world scenarios. For example, if you are calculating the average speed of a vehicle over a distance where time is defined by a quadratic equation, you might encounter rational expressions. Simplifying these allows engineers and scientists to compute values more efficiently and identify critical points where the system might fail (like division by zero). By learning to simplify, you master the ability to reduce complexity in mathematical models.