How many values of $x$ must be excluded in the expression $\frac{x - 2}{(x + 9)(x - 5)}$?
The expression is a rational function given by \(\frac{x-2}{(x+9)(x-5)}\). A rational expression is a fraction where the numerator and denominator are polynomials. Such an expression is undefined whenever the denominator is equal to zero, because division by zero is not defined in mathematics. We need to identify all values of \(x\) that make the denominator \((x+9)(x-5)\) equal to zero.
To find the values of \(x\) that make the denominator zero, we set the expression for the denominator equal to zero:
\[(x+9)(x-5) = 0\]
By the zero product property, a product of factors is zero if and only if at least one of the factors is zero. This gives us two separate equations to solve:
Thus, the expression is undefined at \(x = -9\) and \(x = 5\).
We have found two distinct values of \(x\), specifically \(x = -9\) and \(x = 5\), for which the denominator is zero. Therefore, there are exactly 2 values of \(x\) that must be excluded from the domain of the expression. The final answer is \(\boxed{2}\).
In real-world applications, identifying where a function is undefined is crucial, especially in fields like engineering or physics where functions model physical systems. For example, if you are modeling the voltage in an electrical circuit that depends on a variable resistance \(R\), the formula for the voltage might include a term like \(\frac{1}{R-R_0}\). Knowing that the voltage becomes undefined (or infinite in a physical model) when \(R = R_0\) allows engineers to avoid operating the system at that specific resistance, preventing damage to the circuit components. Understanding domain restrictions is a fundamental skill for ensuring the stability and safety of various technological systems.