To factor the polynomial [tex]\(14x^6 + 8x^3 + 4x^2\)[/tex] completely, we follow these steps:
1. Find the Greatest Common Factor (GCF):
First, identify the GCF of the terms in the polynomial. Here, the terms are [tex]\(14x^6\)[/tex], [tex]\(8x^3\)[/tex], and [tex]\(4x^2\)[/tex].
- The coefficients are [tex]\(14\)[/tex], [tex]\(8\)[/tex], and [tex]\(4\)[/tex]. The GCF of these coefficients is [tex]\(2\)[/tex].
- All the terms have [tex]\(x^2\)[/tex] as a common factor since [tex]\(x^6\)[/tex], [tex]\(x^3\)[/tex], and [tex]\(x^2\)[/tex] all include at least [tex]\(x^2\)[/tex].
Therefore, the GCF of the polynomial is [tex]\(2x^2\)[/tex].
2. Factor out the GCF:
Next, we factor out the GCF from each term in the polynomial:
[tex]\[
14x^6 + 8x^3 + 4x^2 = 2x^2(7x^4) + 2x^2(4x) + 2x^2(2)
\][/tex]
[tex]\[
= 2x^2(7x^4 + 4x + 2)
\][/tex]
3. Verify the factored form:
Finally, we verify that the factored form [tex]\(2x^2(7x^4 + 4x + 2)\)[/tex] is correct by distributing [tex]\(2x^2\)[/tex] back through the polynomial:
[tex]\[
2x^2(7x^4) + 2x^2(4x) + 2x^2(2) = 14x^6 + 8x^3 + 4x^2
\][/tex]
This confirms that the factored form is indeed correct.
Thus, the completely factored form of the polynomial [tex]\(14x^6 + 8x^3 + 4x^2\)[/tex] is:
[tex]\[
2x^2(7x^4 + 4x + 2)
\][/tex]
