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What is the complete factorization of the polynomial below?

[tex]\[ x^3 - 2x^2 + x - 2 \][/tex]

A. [tex]\((x-2)(x+i)(x-i)\)[/tex]

B. [tex]\((x-2)(x-i)(x-i)\)[/tex]

C. [tex]\((x+2)(x-1)(x-1)\)[/tex]

D. [tex]\((x+2)(x+i)(x-1)\)[/tex]


Sagot :

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].