Faelyn grouped the terms and factored the GCF out of the groups of the polynomial 6×4 – 8×2 + 3×2 + 4. Her work is shown.
Step 1: (6×4 – 8×2) + (3×2 + 4)
Step 2: 2×2(3×2 – 4) + 1(3×2 + 4)
Faelyn noticed that she does not have a common factor. Which accurately describes what Faelyn should do next?
Faelyn should realize that her work shows that the polynomial is prime.
Faelyn should go back and regroup the terms in Step 1 as (6×4 + 3×2) – (8×2 + 4).
In Step 2, Faelyn should factor only 2x out of the first expression.
Falyn should factor out a negative from one of the groups so the binomials will be the same.
The Correct Answer and Explanation is :
The correct next step is:
Faelyn should factor out a negative from one of the groups so the binomials will be the same.
Explanation:
Let’s analyze the problem in detail.
Faelyn’s polynomial is:
[
6x^4 – 8x^2 + 3x^2 + 4
]
Step 1: Group the terms
She grouped the polynomial as follows:
[
(6x^4 – 8x^2) + (3x^2 + 4)
]
This is a reasonable first step, but now we need to focus on factoring the greatest common factor (GCF) from each group.
Step 2: Factor out the GCF
Faelyn factored out the GCF from each group:
[
2x^2(3x^2 – 4) + 1(3x^2 + 4)
]
At this stage, there’s an issue. The two binomials ( (3x^2 – 4) ) and ( (3x^2 + 4) ) are not the same, meaning Faelyn cannot combine these terms or factor further unless both binomials match.
What should Faelyn do next?
Faelyn needs to manipulate the terms to make the two binomials identical. The way to achieve this is by factoring out a negative from one of the groups.
Let’s examine this:
[
(6x^4 – 8x^2) + (3x^2 + 4)
]
If we factor out a negative from the first group, we get:
[
-2x^2(3x^2 – 4) + 1(3x^2 + 4)
]
Now, the two binomials are ( (3x^2 – 4) ) and ( (3x^2 + 4) ), which are not the same.
However, the binomial within the parentheses of the second term (3x² + 4) can be factored so that the two binomials match. Since we’re factoring in a negative from the first group, we can group them together.