Question

21
select the correct answer.
what is this expression in simplest form?
\\(\frac{x^2 + x - 2}{x^3 - x^2 + 2x - 2}\\)
a. \\(\frac{1}{x - 2}\\)
b. \\(\frac{1}{x + 2}\\)
c. \\(\frac{x + 2}{x^2 + 2}\\)
d. \\(\frac{x - 1}{x^2 + 2}\\)

Explanation:

Step1: Factor the numerator

Factor \(x^2 + x - 2\). We need two numbers that multiply to \(-2\) and add to \(1\). Those numbers are \(2\) and \(-1\). So, \(x^2 + x - 2=(x + 2)(x - 1)\).

Step2: Factor the denominator

Factor \(x^3 - x^2 + 2x - 2\) by grouping. Group the first two terms and the last two terms:
\((x^3 - x^2)+(2x - 2)=x^2(x - 1)+2(x - 1)=(x - 1)(x^2 + 2)\).

Step3: Simplify the fraction

Now, the original expression \(\frac{x^2 + x - 2}{x^3 - x^2 + 2x - 2}\) becomes \(\frac{(x + 2)(x - 1)}{(x - 1)(x^2 + 2)}\). Cancel out the common factor \((x - 1)\) (assuming \(x
eq1\)) to get \(\frac{x + 2}{x^2 + 2}\).

Answer:

C. \(\frac{x + 2}{x^2 + 2}\)