Select the correct answer.

What is this expression in simplest form?

[tex]\[
\frac{x^2 + x - 2}{x^3 - x^2 + 2x - 2}
\][/tex]

A. [tex]\(\frac{1}{x-2}\)[/tex]

B. [tex]\(\frac{1}{x+2}\)[/tex]

C. [tex]\(\frac{x+2}{x^2+2}\)[/tex]

D. [tex]\(\frac{x-1}{x^2+2}\)[/tex]



Answer :

To determine the simplest form of the given expression

[tex]\[ \frac{x^2 + x - 2}{x^3 - x^2 + 2x - 2}, \][/tex]

we need to simplify the fraction and check which of the provided options it matches. Here is the step-by-step procedure:

1. Factor the Numerator and Denominator:

We start by factoring both the numerator and the denominator if possible.

The numerator [tex]\(x^2 + x - 2\)[/tex] can be factored as:
[tex]\[ x^2 + x - 2 = (x + 2)(x - 1) \][/tex]

The denominator [tex]\(x^3 - x^2 + 2x - 2\)[/tex] can be factored by grouping:
[tex]\[ x^3 - x^2 + 2x - 2 = x^2(x - 1) + 2(x - 1) = (x^2 + 2)(x - 1) \][/tex]

2. Simplify the Expression:

Now we have:
[tex]\[ \frac{(x + 2)(x - 1)}{(x^2 + 2)(x - 1)} \][/tex]

We can cancel the common term [tex]\((x - 1)\)[/tex] in the numerator and the denominator:
[tex]\[ \frac{x + 2}{x^2 + 2} \][/tex]

3. Compare with the Provided Options:

Let's see which option matches [tex]\(\frac{x+2}{x^2+2}\)[/tex]:

- [tex]\( \text{Option A:} \frac{1}{x-2} \)[/tex]
This is not equivalent to [tex]\(\frac{x + 2}{x^2 + 2}\)[/tex].

- [tex]\( \text{Option B:} \frac{1}{x+2} \)[/tex]
This is not equivalent to [tex]\(\frac{x + 2}{x^2 + 2}\)[/tex].

- [tex]\( \text{Option C:} \frac{x+2}{x^2+2} \)[/tex]
This is exactly equivalent to our simplified form [tex]\(\frac{x + 2}{x^2 + 2}\)[/tex].

- [tex]\( \text{Option D:} \frac{x-1}{x^2+2} \)[/tex]
This is not equivalent to [tex]\(\frac{x + 2}{x^2 + 2}\)[/tex].

Thus, the simplest form of the given expression is

[tex]\[ \boxed{ \frac{x+2}{x^2+2} }, \][/tex]

which corresponds to option C.

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