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Factor completely: [tex]14x^6 + 8x^3 + 4x^2[/tex]

A. [tex]2x^2(7x^4 + 4x + 2)[/tex]
B. [tex]2(7x^6 + 4x^3 + 2x^2)[/tex]
C. [tex]x^2(14x^4 + 8x + 4)[/tex]
D. Prime

Sagot :

GinnyAnsweridn
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]
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