- Factor the numerator \(x^2-16x+64\) as \((x-8)^2\).
- Factor the denominator \(x^2-3x-40\) as \((x-8)(x+5)\).
- Cancel the common factor \((x-8)\) to simplify the rational expression.
- The simplified form is \(\boxed{\frac{x-8}{x+5}}\).
Explanation
- Factoring the Numerator and Denominator
The rational expression to simplify is \(\frac{x^2-16x+64}{x^2-3x-40}\). To simplify this into the form \(\frac{x-A}{x+B}\), we must factor both the numerator and the denominator. The numerator is a quadratic trinomial of the form \(x^2-16x+64\), which can be written as \((x-8)^2\). The denominator is a quadratic trinomial of the form \(x^2-3x-40\). We look for two numbers that multiply to \(-40\) and add to \(-3\). These numbers are \(-8\) and \(5\). Thus, the denominator can be factored as \((x-8)(x+5)\).
- Simplifying the Rational Expression
With the expressions factored, we rewrite the rational expression as follows:
\[ \frac{(x-8)^2}{(x-8)(x+5)} \]
We can cancel the common factor of \((x-8)\) from the numerator and the denominator, provided that \(x
eq 8\). This results in:
\[ \frac{x-8}{x+5} \]
This matches the target form \(\frac{x-A}{x+B}\), where \(A = 8\) and \(B = 5\).
- Conclusion and Final Result
By simplifying the expression, we have shown that \(\frac{x^2-16x+64}{x^2-3x-40} = \frac{x-8}{x+5}\). Thus, the values of the missing terms are \(A = 8\) and \(B = 5\), giving us the final result:
\[ \boxed{\frac{x-8}{x+5}} \]
Examples
Simplifying rational expressions is essential in fields like engineering and economics, where complex models often involve relationships that can be reduced to simpler ratios. For example, if you are modeling the efficiency of a machine as it ages, you might represent the output-to-input ratio as a complex polynomial fraction. By simplifying this fraction, you can more easily analyze the trends and identify the specific points at which the system behavior changes, such as the value at which the efficiency factor becomes zero or undefined. Understanding these techniques allows for more efficient problem solving and better design of complex systems.