To factor the polynomial expression [tex]\(2x^3 + 10x^2 + 14x + 70\)[/tex] completely, let's go through the steps:
1. Factor out the Greatest Common Factor (GCF):
First, we observe that each term in the polynomial shares a common factor of 2. So, we factor out the 2:
[tex]\[
2x^3 + 10x^2 + 14x + 70 = 2(x^3 + 5x^2 + 7x + 35)
\][/tex]
2. Check for Further Factoring:
Next, we need to factor the polynomial inside the parentheses: [tex]\(x^3 + 5x^2 + 7x + 35\)[/tex].
3. Identify and group the factors:
One of the common techniques is to look for potential factor groups or patterns in the polynomial. In this case, we notice that the expression [tex]\(x^3 + 5x^2 + 7x + 35\)[/tex] could be factored further by recognizing potential simplifications.
4. Recognize polynomial structure:
It helps to see if the polynomial can be broken into simpler parts. In our polynomial [tex]\(x^3 + 5x^2 + 7x + 35\)[/tex], we realize it can be interpreted through the identification of its factors. By a deeper inspection (or provided computation results), it is confirmed that the polynomial can be factored as:
[tex]\[
x^3 + 5x^2 + 7x + 35 = (x + 5)(x^2 + 7)
\][/tex]
5. Combine with the GCF factored out:
Therefore, combining it with the 2 we factored out earlier, we have:
[tex]\[
2(x^3 + 5x^2 + 7x + 35) = 2 \left((x + 5)(x^2 + 7)\right)
\][/tex]
Thus, the completely factored form of the polynomial [tex]\(2x^3 + 10x^2 + 14x + 70\)[/tex] is:
[tex]\[
2 \left((x + 5)(x^2 + 7)\right)
\][/tex]
So the correct answer is:
[tex]\[
2 \left[\left(x^2 + 7\right)(x + 5)\right]
\][/tex]
