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

Step 1: $\left(6 x^4-8 x^2\right)+\left(3 x^2+4\right)$
Step 2: $2 x^2\left(3 x^2-4\right)+1\left(3 x^2+4\right)$

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 $\left(6 x^4+3 x^2\right)-\left(8 x^2+4\right)$.
C. In Step 2, Faelyn should factor only $2 x$ out of the first expression.
D. Faelyn should factor out a negative from one of the groups so the binomials will be the same.

Faelyn should regroup the polynomial as ( 6 x 4 + 3 x 2 ) − ( 8 x 2 + 4 ) to find a common factor. This allows her to factor out ( 2 x 2 + 1 ) and complete the factoring process. Therefore, the correct choice is option B.
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Answered by Anonymous | 2025-07-04

Regroup the terms of the polynomial as ( 6 x 4 + 3 x 2 ) − ( 8 x 2 + 4 ) .
Factor out the GCF from each group: 3 x 2 ( 2 x 2 + 1 ) − 4 ( 2 x 2 + 1 ) .
Factor out the common binomial: ( 2 x 2 + 1 ) ( 3 x 2 − 4 ) .
The correct next step is to regroup the terms as described. Faelyn should go back and regroup the terms in Step 1 as ( 6 x 4 + 3 x 2 ) − ( 8 x 2 + 4 ) . ​

Explanation

Analyze the Problem The problem is to identify the correct next step for Faelyn to factor the polynomial 6 x 4 − 8 x 2 + 3 x 2 + 4 given her initial steps. She grouped the terms as ( 6 x 4 − 8 x 2 ) + ( 3 x 2 + 4 ) and factored out the greatest common factor (GCF) from each group, resulting in 2 x 2 ( 3 x 2 − 4 ) + 1 ( 3 x 2 + 4 ) . She noticed that she does not have a common factor. We need to determine the correct action for Faelyn to proceed.

Evaluate the Options Let's analyze the given options:

Faelyn should realize that her work shows that the polynomial is prime. This might be true, but we should first explore other factoring possibilities before concluding that the polynomial is prime.

Faelyn should go back and regroup the terms in Step 1 as ( 6 x 4 + 3 x 2 ) − ( 8 x 2 + 4 ) . Let's try this regrouping: ( 6 x 4 + 3 x 2 ) − ( 8 x 2 + 4 ) = 3 x 2 ( 2 x 2 + 1 ) − 4 ( 2 x 2 + 1 ) . This gives us a common factor of ( 2 x 2 + 1 ) .

In Step 2, Faelyn should factor only 2 x out of the first expression. If she factors out 2 x , she would have 2 x ( 3 x 3 − 4 x ) + 1 ( 3 x 2 + 4 ) , which does not lead to a common factor.

Faelyn should factor out a negative from one of the groups so the binomials will be the same. This is not directly applicable to the current expression.

Perform Regrouping and Factoring From the analysis, regrouping the terms as ( 6 x 4 + 3 x 2 ) − ( 8 x 2 + 4 ) leads to a common factor. Factoring out the GCF from each group gives us:


3 x 2 ( 2 x 2 + 1 ) − 4 ( 2 x 2 + 1 )

Factor out the Common Binomial Now, we can factor out the common binomial factor ( 2 x 2 + 1 ) :

( 2 x 2 + 1 ) ( 3 x 2 − 4 )

Conclusion Therefore, the correct next step for Faelyn is to go back and regroup the terms in Step 1 as ( 6 x 4 + 3 x 2 ) − ( 8 x 2 + 4 ) .

Examples
Factoring polynomials is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use factoring to simplify complex equations when designing structures or circuits. Similarly, economists use factoring to analyze market trends and predict future economic behavior. By mastering factoring techniques, students can develop problem-solving skills that are applicable to a wide range of fields.

Answered by GinnyAnswer | 2025-07-04