- Factor the numerator \(x^2 + x - 2\) as \((x + 2)(x - 1)\).
- Factor the denominator \(x^3 - x^2 + 2x - 2\) by grouping as \((x^2 + 2)(x - 1)\).
- Cancel the common factor \((x - 1)\) from the numerator and denominator to simplify the expression.
- The final simplified expression is \(\boxed{\frac{x+2}{x^2+2}}\).
Explanation
- Analysis and Objectives
The goal is to simplify the rational expression \(\frac{x^2+x-2}{x^3-x^2+2x-2}\). To do this, we need to factor both the numerator and the denominator completely and identify any common factors. A common factor can be canceled from both, simplifying the expression significantly.
- Factoring the Numerator
First, consider the numerator \(x^2 + x - 2\). We seek two numbers that multiply to \(-2\) and add up to \(1\). These numbers are \(2\) and \(-1\). Thus, we can factor the numerator as:
\[x^2 + x - 2 = (x + 2)(x - 1)\]
- Factoring the Denominator
Next, consider the denominator \(x^3 - x^2 + 2x - 2\). This is a polynomial with four terms, so we can attempt to factor it by grouping:
\[(x^3 - x^2) + (2x - 2) = x^2(x - 1) + 2(x - 1)\]
Notice that both terms share a common factor of \((x - 1)\). Factoring out \((x - 1)\), we get:
\[(x^2 + 2)(x - 1)\]
- Simplifying the Expression
Now, combine the factored numerator and denominator:
\[\frac{(x + 2)(x - 1)}{(x^2 + 2)(x - 1)}\]
Assuming \(x
eq 1\), we can cancel the common factor \((x - 1)\) from both the numerator and the denominator. This leaves us with the simplified expression:
\[\frac{x + 2}{x^2 + 2}\]
- Conclusion
Comparing our result with the provided options, we find that \(\frac{x+2}{x^2+2}\) matches the third option. Therefore, the simplified form is \(\boxed{\frac{x+2}{x^2+2}}\).
Examples
Simplifying rational expressions is a fundamental skill in algebra, useful for solving complex equations and analyzing function behavior. For instance, if you are designing a digital filter, you might deal with a transfer function in the form of a ratio of polynomials. Reducing this ratio to its simplest form can reveal critical information about the filter's stability and frequency response without the computational overhead of dealing with higher-degree polynomials. By factoring out common components, you make complex systems more manageable and easier to interpret in real-world engineering or data analysis scenarios.