To determine the factors of the polynomial [tex]\(6x^3 + 6\)[/tex], we can begin by observing that the polynomial can be factored in several steps:
1. Identify the common factor: The polynomial [tex]\(6x^3 + 6\)[/tex] has a common factor of [tex]\(6\)[/tex].
[tex]\(6x^3 + 6 = 6(x^3 + 1)\)[/tex]
2. Recognize the sum of cubes: Notice that [tex]\(x^3 + 1\)[/tex] is a sum of cubes, which can be factored using the formula [tex]\(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)[/tex]. For [tex]\(x^3 + 1\)[/tex], we have:
[tex]\[
x^3 + 1 = x^3 + 1^3 = (x + 1)(x^2 - x \cdot 1 + 1^2)
\][/tex]
Simplifying, we get:
[tex]\[
x^3 + 1 = (x + 1)(x^2 - x + 1)
\][/tex]
3. Combine the factors: Now we can substitute back into the original expression:
[tex]\[
6(x^3 + 1) = 6((x + 1)(x^2 - x + 1))
\][/tex]
4. Factor representation: Thus, the polynomial [tex]\(6x^3 + 6\)[/tex] can be expressed in its factored form as:
[tex]\[
6x^3 + 6 = 6(x + 1)(x^2 - x + 1)
\][/tex]
Therefore, the factors of [tex]\(6x^3 + 6\)[/tex] are:
- [tex]\(6\)[/tex]
- [tex]\((x + 1)\)[/tex]
- [tex]\((x^2 - x + 1)\)[/tex]
Given the options, if we are to identify which of these is a factor, we see that [tex]\((x^2 - x + 1)\)[/tex] and [tex]\((x + 1)\)[/tex] appear explicitly as factors. Additionally, the constant [tex]\(6\)[/tex] is also a factor in the product. So, any of these would be correct.
Thus, each of the following components is a factor: [tex]\(6\)[/tex], [tex]\((x + 1)\)[/tex], or [tex]\((x^2 - x + 1)\)[/tex].
