Simplify: $\frac{x^2-16}{x-4}$
We start by examining the rational expression \(\frac{x^2 - 16}{x - 4}\). A rational expression is undefined whenever its denominator is equal to zero because division by zero is not possible. To identify the excluded values, we set the denominator equal to zero and solve for \(x\):
\[x - 4 = 0 \implies x = 4.\]
Thus, the value \(x = 4\) is excluded from the domain of the expression.
Next, we look to simplify the expression by factoring the numerator. The numerator \(x^2 - 16\) is a classic difference of squares, which follows the algebraic identity \(a^2 - b^2 = (a - b)(a + b)\). Setting \(a = x\) and \(b = 4\), we have:
\[x^2 - 16 = x^2 - 4^2 = (x - 4)(x + 4).\]
Now, we can rewrite the entire expression as:
\[\frac{x^2 - 16}{x - 4} = \frac{(x - 4)(x + 4)}{x - 4}.\]
With the expression rewritten, we can cancel the common factor \((x - 4)\) from both the numerator and the denominator, provided that \(x
eq 4\):
\[\frac{(x - 4)(x + 4)}{x - 4} = x + 4, \quad \text{for } x
eq 4.\]
The expression simplifies to \(x + 4\).
Combining the steps above, the expression \(\frac{x^2 - 16}{x - 4}\) simplifies to \(x + 4\) with the condition that \(x eq 4\). Therefore, the excluded value is \(4\), and the simplified form is \(x + 4\).
In engineering or physics, simplified expressions are often used to model movement. For instance, if the velocity of an object is given by a complex rational expression that models resistance or acceleration, simplifying it can make calculations significantly faster. Imagine a scenario where a system's efficiency is modeled by \(\frac{x^2 - 16}{x - 4}\); calculating its behavior at various values becomes much easier once we see it is simply \(x+4\), as long as we remember to avoid the singularity at \(x=4\) where the model breaks down.