To determine the factors of the polynomial [tex]\(x^3-12x^2-2x+24\)[/tex] by grouping, let's follow these steps:
1. Group the terms in pairs:
[tex]\[
x^3 - 12x^2 - 2x + 24 = (x^3 - 12x^2) + (-2x + 24)
\][/tex]
2. Factor out the greatest common factor (GCF) from each pair:
For the first pair [tex]\(x^3 - 12x^2\)[/tex]:
[tex]\[
x^3 - 12x^2 = x^2(x - 12)
\][/tex]
For the second pair [tex]\(-2x + 24\)[/tex]:
[tex]\[
-2x + 24 = -2(x - 12)
\][/tex]
So, the expression now looks like:
[tex]\[
x^3 - 12x^2 - 2x + 24 = x^2(x - 12) - 2(x - 12)
\][/tex]
3. Factor out the common binomial factor [tex]\((x - 12)\)[/tex]:
[tex]\[
x^2(x - 12) - 2(x - 12) = (x^2 - 2)(x - 12)
\][/tex]
Hence, the factorization of the polynomial [tex]\( x^3 - 12x^2 - 2x + 24 \)[/tex] by grouping is:
[tex]\[
\boxed{x^2(x - 12) - 2(x - 12)}
\][/tex]
So, the correct option that shows one way to determine the factors of [tex]\( x^3-12x^2-2x+24 \)[/tex] by grouping is:
[tex]\[
x^2(x-12)-2(x-12)
\][/tex]
