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Faelyn grouped the terms and factored the GCF out of the groups of the polynomial [tex]$6x^4 - 8x^2 + 3x^2 + 4$[/tex]. Her work is shown below:

Step 1: [tex]$(6x^4 - 8x^2) + (3x^2 + 4)$[/tex]
Step 2: [tex][tex]$2x^2(3x^2 - 4) + 1(3x^2 + 4)$[/tex][/tex]

Faelyn noticed that she does not have a common factor. Which accurately describes what Faelyn should do next?

A. Faelyn should realize that her work shows that the polynomial is prime.
B. Faelyn should go back and regroup the terms in Step 1 as [tex]$(6x^4 + 3x^2) - (8x^2 + 4)$[/tex].
C. In Step 2, Faelyn should factor only [tex]$2x$[/tex] out of the first expression.
D. Faelyn should factor out a negative from one of the groups so the binomials will be the same.

Sagot :

GinnyAnswer
Let's go through the steps in detail to address what Faelyn should do next.

1. Original Grouping:
Given polynomial: [tex]\(6x^4 - 8x^2 + 3x^2 + 4\)[/tex]

Grouping the terms:
[tex]\[ (6x^4 - 8x^2) + (3x^2 + 4) \][/tex]

2. Factoring Out the GCF in Each Group:
Factor out the greatest common factor (GCF) from each grouped term:
[tex]\[ 2x^2(3x^2 - 4) + 1(3x^2 + 4) \][/tex]
At this stage, Faelyn sees that there is no common binomial factor between the two grouped expressions, i.e., [tex]\(3x^2 - 4\)[/tex] and [tex]\(3x^2 + 4\)[/tex].

3. Regrouping the Terms:
To make the binomial factors the same, Faelyn should factor out a negative sign from one of the groups. This will align the binomials properly:
[tex]\[ (6x^4 - 8x^2) - (-(3x^2 + 4)) \][/tex]

4. Factoring Out the GCF Again:
By factoring a negative from the second group, we can now express it as:
[tex]\[ 6x^4 - 8x^2 - (3x^2 + 4) \][/tex]
Simplifying the second group:
[tex]\[ (6x^4 - 8x^2) - (-3x^2 - 4) \][/tex]

Now factor out the common binomial factor from the expression:
[tex]\[ 2x^2(3x^2 - 4) - 1(3x^2 - 4) \][/tex]

5. Combining the Factored Terms:
With the same binomial factor [tex]\(3x^2 - 4\)[/tex] in place, we can combine the factored terms:
[tex]\[ (3x^2 - 4)(2x^2 - 1) \][/tex]

Thus, the polynomial [tex]\(6x^4 - 8x^2 + 3x^2 + 4\)[/tex] can be factored as [tex]\((3x^2 - 4)(2x^2 - 1)\)[/tex].

Hence, the accurate description of what Faelyn should do next is:
Faelyn should factor out a negative from one of the groups so the binomials will be the same.
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