Let's go through the steps to determine the factors of the polynomial [tex]\( x^3 - 12x^2 - 2x + 24 \)[/tex] by grouping:
1. Consider the polynomial: [tex]\( x^3 - 12x^2 - 2x + 24 \)[/tex].
2. Group the terms in pairs:
- Group the first two terms and the last two terms: [tex]\( (x^3 - 12x^2) + (-2x + 24) \)[/tex].
3. Factor out the common terms in each group:
- From the first group [tex]\( (x^3 - 12x^2) \)[/tex], factor out [tex]\( x^2 \)[/tex]:
[tex]\[ x^2(x - 12) \][/tex]
- From the second group [tex]\( (-2x + 24) \)[/tex], factor out [tex]\(-2\)[/tex]:
[tex]\[ -2(x - 12) \][/tex]
4. Combine the factored groups:
- After factoring, we write the polynomial as:
[tex]\[ x^2(x - 12) - 2(x - 12) \][/tex]
5. Factor out the common term [tex]\((x - 12)\)[/tex]:
- Notice that [tex]\((x - 12)\)[/tex] is a common factor in both terms:
[tex]\[ (x^2 - 2)(x - 12) \][/tex]
From the above steps, the correct grouping that shows one way to determine the factors of [tex]\( x^3 - 12x^2 - 2x + 24 \)[/tex] by grouping is:
[tex]\[ \boxed{x^2(x - 12) - 2(x - 12)} \][/tex]
