- Factor the denominator of the first term: \(x^2 - 16 = (x-4)(x+4)\).
- Use this as a common denominator to rewrite the second term: \(\frac{3(x+4)}{(x-4)(x+4)}\).
- Combine and simplify the numerators: \(x - 3(x+4) = x - 3x - 12 = -2x - 12 = -2(x+6)\).
- Resulting in the simplified expression: \(\boxed{\frac{-2(x+6)}{(x+4)(x-4)}}\).
Explanation
- Analyze the Denominators
To simplify the expression \(\frac{x}{x^2-16} - \frac{3}{x-4}\), we first analyze the denominators. Notice that the denominator of the first fraction, \(x^2-16\), is a difference of squares. We can factor this as:
\[x^2 - 16 = (x - 4)(x + 4)\]
Since this factor includes the denominator of the second fraction, \((x-4)\), we can use \((x-4)(x+4)\) as the common denominator. This requires that \(x
eq 4\) and \(x
eq -4\) to avoid division by zero.
- Find a Common Denominator
We now rewrite the second fraction, \(\frac{3}{x-4}\), to have the common denominator \((x-4)(x+4)\). To do this, we multiply both the numerator and the denominator by \((x+4)\):
\[\frac{3}{x-4} = \frac{3(x+4)}{(x-4)(x+4)} = \frac{3x+12}{(x-4)(x+4)}\]
Now we can express the original subtraction problem with a common denominator:
\[\frac{x}{(x-4)(x+4)} - \frac{3(x+4)}{(x-4)(x+4)}\]
- Perform the Subtraction and Simplify the Numerator
With a common denominator, we can combine the numerators:
\[\frac{x - 3(x+4)}{(x-4)(x+4)}\]
Next, distribute the \(-3\) in the numerator:
\[x - 3x - 12\]
Combining like terms (\(x - 3x = -2x\)), we get:
\[-2x - 12\]
We can then factor out \(-2\) from this expression:
\[-2(x + 6)\]
So the entire fraction simplifies to:
\[\frac{-2(x+6)}{(x-4)(x+4)}\]
- Final Conclusion
Comparing our simplified expression \(\frac{-2(x+6)}{(x-4)(x+4)}\) with the given options, we identify that it matches the second option. Thus, the correct difference is \(\boxed{\frac{-2(x+6)}{(x+4)(x-4)}}\).
Examples
When engineers design mechanical systems or electrical circuits, they often encounter complex rational expressions that represent impedance or transfer functions. Simplifying these expressions is a fundamental skill in solving for system stability or output responses. For example, if two subsystems have output characteristics represented by \(\frac{x}{x^2-16}\) and \(\frac{3}{x-4}\), finding their difference helps in analyzing the system's differential performance or identifying resonant frequencies where denominators might vanish. Simplifying these rational functions ensures that the model is manageable and ready for further algebraic or numerical analysis.