Factor completely $2 x^3+8 x^2+3 x+12$
A. $\left(2 x^2+3\right)(x+4)$
B. $\left(2 x^2+3\right)(x-4)$
C. $\left(2 x^2-3\right)(x+4)$
D. $\left(2 x^2-3\right)(x-4)$
Answer (2)
Group the terms: ( 2 x 3 + 8 x 2 ) + ( 3 x + 12 ) .
Factor out the GCF from each group: 2 x 2 ( x + 4 ) + 3 ( x + 4 ) .
Factor out the common binomial: ( 2 x 2 + 3 ) ( x + 4 ) .
The complete factorization is ( 2 x 2 + 3 ) ( x + 4 ) .
Explanation
Problem Analysis We are given the polynomial 2 x 3 + 8 x 2 + 3 x + 12 and asked to factor it completely. We will use factoring by grouping to achieve this.
Grouping Terms First, group the terms as follows: ( 2 x 3 + 8 x 2 ) + ( 3 x + 12 ) .
Factoring out GCF Next, factor out the greatest common factor (GCF) from each group. From the first group, 2 x 2 can be factored out, and from the second group, 3 can be factored out: 2 x 2 ( x + 4 ) + 3 ( x + 4 ) .
Factoring out the Common Binomial Now, we can see that ( x + 4 ) is a common binomial factor. Factor it out: ( 2 x 2 + 3 ) ( x + 4 ) .
Checking for Further Factorization The quadratic 2 x 2 + 3 cannot be factored further using real numbers because if we set 2 x 2 + 3 = 0 , we get x 2 = − 2 3 , which has no real solutions. Therefore, the complete factorization is ( 2 x 2 + 3 ) ( x + 4 ) .
Final Factorization Thus, the completely factored form of the given polynomial is ( 2 x 2 + 3 ) ( x + 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 supply and demand curves. Factoring helps to break down complex problems into simpler, more manageable parts, making it easier to find solutions and make informed decisions. Understanding factoring can also help in optimizing resources and predicting outcomes in various fields.
The polynomial 2 x 3 + 8 x 2 + 3 x + 12 factors completely to ( 2 x 2 + 3 ) ( x + 4 ) . Thus, the correct option is A. This was achieved through factoring by grouping and extracting common factors.
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