Factor completely $14 b x^2-7 x^3-4 b+2 x$.

A. $(2 b+x)(7 x^2+2)$
B. $(2 b-x)(7 x^2-2)$
C. $(2 b+x)(7 x^2-2)$
D. Prime

Rearrange the terms: 14 b x 2 − 4 b − 7 x 3 + 2 x .
Factor out common factors: 2 b ( 7 x 2 − 2 ) − x ( 7 x 2 − 2 ) .
Factor out the common binomial: ( 2 b − x ) ( 7 x 2 − 2 ) .
The completely factored expression is ( 2 b − x ) ( 7 x 2 − 2 ) ​ .

Explanation

Understanding the Problem We are given the expression 14 b x 2 − 7 x 3 − 4 b + 2 x and asked to factor it completely.

Rearranging Terms First, let's rearrange the terms to group similar terms together: 14 b x 2 − 7 x 3 − 4 b + 2 x = 14 b x 2 − 4 b − 7 x 3 + 2 x

Factoring Common Factors Next, we factor out common factors from the grouped terms: 14 b x 2 − 4 b − 7 x 3 + 2 x = 2 b ( 7 x 2 − 2 ) − x ( 7 x 2 − 2 )

Factoring the Common Binomial Now, we factor out the common binomial factor ( 7 x 2 − 2 ) :
2 b ( 7 x 2 − 2 ) − x ( 7 x 2 − 2 ) = ( 2 b − x ) ( 7 x 2 − 2 )

Final Factorization Finally, we check if the factors can be further factored. In this case, they cannot be factored further. Therefore, the completely factored expression is ( 2 b − x ) ( 7 x 2 − 2 ) .


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 analyzing circuits. In economics, factoring can be used to analyze cost and revenue functions to determine break-even points. In computer graphics, factoring can be used to optimize rendering algorithms.

Answered by GinnyAnswer | 2025-07-03

The expression 14 b x 2 − 7 x 3 − 4 b + 2 x factors to ( 2 b − x ) ( 7 x 2 − 2 ) . This is achieved by rearranging, grouping, factoring out common terms, and recognizing common binomial factors. Hence, the answer corresponds to option B.
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Answered by Anonymous | 2025-07-04