To factor the polynomial [tex]\( x^3 - 2x^2 + x - 2 \)[/tex] completely, we need to follow several steps. Let's start by finding its roots.
### Step 1: Rational Root Theorem
The Rational Root Theorem states that any rational root of a polynomial equation, say [tex]\( p(x) = 0 \)[/tex], is a fraction [tex]\( \frac{p}{q} \)[/tex], where [tex]\( p \)[/tex] is a factor of the constant term and [tex]\( q \)[/tex] is a factor of the leading coefficient.
For the polynomial [tex]\( x^3 - 2x^2 + x - 2 \)[/tex]:
- The constant term is [tex]\(-2\)[/tex].
- The leading coefficient is [tex]\(1\)[/tex].
The possible rational roots are [tex]\( \pm 1, \pm 2 \)[/tex].
### Step 2: Test the Potential Rational Roots
We will now test these potential roots by substituting them into the polynomial and checking if it equals zero.
#### Testing [tex]\( x = 1 \)[/tex]:
[tex]\[ 1^3 - 2(1)^2 + 1 - 2 = 1 - 2 + 1 - 2 = -2 \][/tex]
[tex]\[ \Rightarrow x = 1 \text{ is not a root.} \][/tex]
#### Testing [tex]\( x = -1 \)[/tex]:
[tex]\[ (-1)^3 - 2(-1)^2 + (-1) - 2 = -1 - 2 - 1 - 2 = -6 \][/tex]
[tex]\[ \Rightarrow x = -1 \text{ is not a root.} \][/tex]
#### Testing [tex]\( x = 2 \)[/tex]:
[tex]\[ 2^3 - 2(2)^2 + 2 - 2 = 8 - 8 + 2 - 2 = 0 \][/tex]
[tex]\[ \Rightarrow x = 2 \text{ is a root.} \][/tex]
### Step 3: Polynomial Division
Since [tex]\( x = 2 \)[/tex] is a root, we know that [tex]\( x - 2 \)[/tex] is a factor of the polynomial. We will now divide [tex]\( x^3 - 2x^2 + x - 2 \)[/tex] by [tex]\( x - 2 \)[/tex] using synthetic division or long division.
#### Synthetic Division:
[tex]\[
\begin{array}{r|rrrr}
2 & 1 & -2 & 1 & -2 \\
\hline
& & 2 & 0 & 2 \\
\hline
& 1 & 0 & 1 & 0 \\
\end{array}
\][/tex]
The result of the division is [tex]\( x^2 + 1 \)[/tex], so:
[tex]\[
x^3 - 2x^2 + x - 2 = (x - 2)(x^2 + 1)
\][/tex]
### Step 4: Factor [tex]\( x^2 + 1 \)[/tex]
Notice that [tex]\( x^2 + 1 \)[/tex] can be factored further over the complex numbers:
[tex]\[
x^2 + 1 = (x + i)(x - i)
\][/tex]
### Conclusion
Putting it all together, the complete factorization of [tex]\( x^3 - 2x^2 + x - 2 \)[/tex] over the complex numbers is:
[tex]\[
x^3 - 2x^2 + x - 2 = (x - 2)(x + i)(x - i)
\][/tex]
Thus, the correct answer is [tex]\( \boxed{A.(x-2)(x+i)(x-i)} \)[/tex].
