giannisdagoat
Answered

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]\( 2x^2(3x^2 - 4) + 1(3x^2 + 4) \)[/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.



Answer :

GinnyAnswer
Let's analyze Faelyn's work step by step.

Faelyn's original polynomial is:
[tex]\[6 x^4 - 8 x^2 + 3 x^2 + 4\][/tex]

Step 1:
Faelyn grouped the terms as:
[tex]\[\left(6 x^4 - 8 x^2\right) + \left(3 x^2 + 4\right)\][/tex]

Step 2:
She factored the GCF out of the groups:
[tex]\[2 x^2 (3 x^2 - 4) + 1 (3 x^2 + 4)\][/tex]

Here Faelyn noticed she doesn't have a common factor between the groups.

Let's identify where Faelyn should go next.

Upon closer inspection, Faelyn should go back and regroup the terms in Step 1:
[tex]\[\left(6 x^4 + 3 x^2\right) - \left(8 x^2 + 4\right)\][/tex]

Next, factor the GCF out of each group:

Regrouped:
[tex]\[\left(6 x^4 + 3 x^2\right) - \left(8 x^2 + 4\right)\][/tex]

Factor out the greatest common factors from each group:
[tex]\[3 x^2 \left(2 x^2 + 1\right) - 4 \left(2 x^2 + 1\right)\][/tex]

Now, we see that [tex]\((2 x^2 + 1)\)[/tex] is a common factor. So, we can factor [tex]\((2 x^2 + 1)\)[/tex] out:
[tex]\[\left(2 x^2 + 1\right)\left(3 x^2 - 4\right)\][/tex]

This indicates what Faelyn should have done:

The accurate action for Faelyn:
Faelyn should go back and regroup the terms in Step 1 as [tex]\(\left(6 x^4 + 3 x^2\right) - \left(8 x^2 + 4\right)\)[/tex].

Therefore, the correct choice is:

Faelyn should go back and regroup the terms in Step 1 as [tex]\(\left(6 x^4 + 3 x^2\right) - \left(8 x^2 + 4\right)\)[/tex].