To determine a factor of the polynomial [tex]\(3x^3 + 18x^2 + 27x\)[/tex], let's go through the steps to factor it step-by-step.
1. Identify the Greatest Common Factor (GCF):
Notice that each term in the polynomial [tex]\(3x^3 + 18x^2 + 27x\)[/tex] has a common factor of [tex]\(3x\)[/tex].
[tex]\[
3x^3 + 18x^2 + 27x = 3x(x^2 + 6x + 9)
\][/tex]
2. Factor the Quadratic Expression:
Next, factor the quadratic expression [tex]\(x^2 + 6x + 9\)[/tex]. This expression is a perfect square trinomial, which can be factored as:
[tex]\[
x^2 + 6x + 9 = (x + 3)^2
\][/tex]
3. Combine the Factors:
Putting it all together, we have:
[tex]\[
3x(x^2 + 6x + 9) = 3x(x + 3)^2
\][/tex]
Thus, the fully factored form of the polynomial [tex]\(3x^3 + 18x^2 + 27x\)[/tex] is:
[tex]\[
3x(x + 3)^2
\][/tex]
From the above factored form, we see the factors are [tex]\(3\)[/tex], [tex]\(x\)[/tex], and [tex]\((x + 3)\)[/tex].
To directly answer the question: Which of the following is a factor of [tex]\(3x^3 + 18x^2 + 27x\)[/tex]?
The factors are [tex]\(x\)[/tex] and [tex]\((x + 3)\)[/tex].
Explicitly, one of the factors is:
[tex]\[
x + 3
\][/tex]
Another factor is simply:
[tex]\[
x
\][/tex]
These two factors satisfy the requirements for being a factor of the given polynomial.
