To factorize the polynomial [tex]\( x^3 - 2x^2 - x + 2 \)[/tex], we follow these steps:
1. Identify possible rational roots: By the Rational Root Theorem, the possible rational roots of the polynomial are the factors of the constant term (2) divided by the factors of the leading coefficient (1). So the possible rational roots are [tex]\( \pm 1, \pm 2 \)[/tex].
2. Test the possible roots: We substitute these possible roots into the polynomial to determine if they are indeed roots.
- For [tex]\( x = 1 \)[/tex]:
[tex]\[
1^3 - 2(1)^2 - 1 + 2 = 1 - 2 - 1 + 2 = 0
\][/tex]
So, [tex]\( x = 1 \)[/tex] is a root of the polynomial.
3. Factor out [tex]\( x - 1 \)[/tex]: Since [tex]\( x = 1 \)[/tex] is a root, [tex]\( (x - 1) \)[/tex] is a factor of the polynomial. Let's use polynomial long division or synthetic division to factor out [tex]\( (x - 1) \)[/tex] from [tex]\( x^3 - 2x^2 - x + 2 \)[/tex].
Performing synthetic division with [tex]\( x = 1 \)[/tex]:
[tex]\[
\begin{array}{r|rrrr}
1 & 1 & -2 & -1 & 2 \\
\hline
& 1 & -1 & -2 & 0 \\
\end{array}
\][/tex]
The coefficients of the quotient polynomial are [tex]\( 1, -1, -2 \)[/tex], giving us [tex]\( x^2 - x - 2 \)[/tex].
4. Factor the quadratic polynomial: Next, we need to factorize [tex]\( x^2 - x - 2 \)[/tex].
The quadratic [tex]\( x^2 - x - 2 \)[/tex] can be factored by finding two numbers that multiply to [tex]\(-2\)[/tex] and add to [tex]\(-1\)[/tex]. These numbers are [tex]\(-2\)[/tex] and [tex]\(1\)[/tex].
Therefore:
[tex]\[
x^2 - x - 2 = (x - 2)(x + 1)
\][/tex]
5. Combine the factors: Now, we combine all the factors we've found:
[tex]\[
x^3 - 2x^2 - x + 2 = (x - 1)(x - 2)(x + 1)
\][/tex]
So, the factorization of the polynomial [tex]\( x^3 - 2x^2 - x + 2 \)[/tex] is:
[tex]\[
(x - 1)(x - 2)(x + 1)
\][/tex]
