To find the completely factored form of the polynomial [tex]\( f(x) = x^3 - 2x^2 - 5x + 6 \)[/tex], we need to factor it into a product of simpler polynomials.
The completely factored form of [tex]\( f(x) = x^3 - 2x^2 - 5x + 6 \)[/tex] is:
[tex]\[
f(x) = (x - 3)(x - 1)(x + 2).
\][/tex]
This process involves identifying the roots of the polynomial and expressing [tex]\( f(x) \)[/tex] as a product of linear polynomials corresponding to those roots. Let's map the given polynomial to its factored equivalents step-by-step:
1. Identify the possible rational roots based on the Rational Root Theorem, which suggests the possible rational roots are the factors of the constant term (6) divided by the factors of the leading coefficient (1). The possible candidates are [tex]\(\pm 1, \pm 2, \pm 3, \pm 6\)[/tex].
2. Test these values to find the actual roots.
3. Once the roots have been determined, we can express the polynomial as a product of its factors in the form [tex]\((x - r_1)(x - r_2)(x - r_3)\)[/tex], where [tex]\(r_1, r_2,\)[/tex] and [tex]\(r_3\)[/tex] are the roots.
The correctly factored form of the given polynomial is:
[tex]\[
f(x) = (x - 3)(x - 1)(x + 2).
\][/tex]
So the correct answer from the choices provided is:
[tex]\[
f(x) = (x + 2)(x - 3)(x - 1).
\][/tex]
