To determine the factors of the polynomial [tex]\( x^3 - 12x^2 - 2x + 24 \)[/tex] by grouping, let's follow a step-by-step process.
### Step 1: Group the Terms
Group the polynomial into pairs of terms:
[tex]\[ x^3 - 12x^2 - 2x + 24 = (x^3 - 12x^2) + (-2x + 24) \][/tex]
### Step 2: Factor out the Greatest Common Factor (GCF) from Each Pair
For the first group [tex]\(x^3 - 12x^2\)[/tex]:
[tex]\[ x^3 - 12x^2 = x^2(x - 12) \][/tex]
For the second group [tex]\(-2x + 24\)[/tex]:
[tex]\[ -2x + 24 = -2(x - 12) \][/tex]
### Step 3: Factor Out the Common Binomial Factor
Now our expression looks like this:
[tex]\[ x^2(x - 12) - 2(x - 12) \][/tex]
Notice that [tex]\(x - 12\)[/tex] is a common factor in both terms. We can factor it out:
[tex]\[ x^2(x - 12) - 2(x - 12) = (x^2 - 2)(x - 12) \][/tex]
### Conclusion
Thus, the polynomial [tex]\( x^3 - 12x^2 - 2x + 24 \)[/tex] can be factored as:
[tex]\[ (x^2 - 2)(x - 12) \][/tex]
Among the given options, the correct step that shows one way to determine the factors of the polynomial by grouping is:
[tex]\[ x^2(x - 12) - 2(x - 12) \][/tex]
