To fully factorize the polynomial [tex]\( f(x) = x^3 + 2x^2 - x - 2 \)[/tex] by grouping and extracting the highest common multiple, we can follow these steps:
1. Group the terms in pairs to facilitate the factoring process by grouping:
[tex]\[
f(x) = x^3 + 2x^2 - x - 2 = (x^3 + 2x^2) + (-x - 2)
\][/tex]
2. Factor out the greatest common factor (GCF) from each pair:
[tex]\[
= x^2(x + 2) - 1(x + 2)
\][/tex]
3. Now, factor out the common binomial factor [tex]\((x + 2)\)[/tex] from the grouped expression:
[tex]\[
= (x^2 - 1)(x + 2)
\][/tex]
4. Notice that [tex]\( x^2 - 1 \)[/tex] is a difference of squares:
[tex]\[
x^2 - 1 = (x - 1)(x + 1)
\][/tex]
5. Hence, substitute the factored form of [tex]\( x^2 - 1 \)[/tex] back into the expression:
[tex]\[
f(x) = (x - 1)(x + 1)(x + 2)
\][/tex]
Therefore, the fully factorized form of the polynomial [tex]\( f(x) = x^3 + 2x^2 - x - 2 \)[/tex] is:
[tex]\[
f(x) = (x - 1)(x + 1)(x + 2)
\][/tex]
