Louise begins factoring the polynomial, which has four terms:
[tex]
\begin{array}{l}
8 x^3+24 x^2+10 x+30 \\
2\left(4 x^3+12 x^2+5 x+15\right) \\
2\left[4 x^2(x+3)+5(x+3)\right]
\end{array}
[/tex]
Which is the completely factored form of her polynomial?
A. [tex]2 x\left(4 x^2+5\right)[/tex]
B. [tex]2\left(4 x^2+6\right)(x+3)[/tex]
C. [tex]2\left(4 x^2+5\right)(x+3)[/tex]
D. [tex]2\left(4 x^2+5\right)(x+3)[/tex]
Answer (2)
Factor out the common factor 2: 8 x 3 + 24 x 2 + 10 x + 30 = 2 ( 4 x 3 + 12 x 2 + 5 x + 15 ) .
Group terms and factor: 2 [ 4 x 2 ( x + 3 ) + 5 ( x + 3 )] .
Factor out the common binomial ( x + 3 ) : 2 ( 4 x 2 + 5 ) ( x + 3 ) .
The completely factored form is 2 ( 4 x 2 + 5 ) ( x + 3 ) .
Explanation
Analyzing the Problem We are given the polynomial 8 x 3 + 24 x 2 + 10 x + 30 and Louise has started factoring it as 2 [ 4 x 2 ( x + 3 ) + 5 ( x + 3 )] . We need to complete the factorization.
Factoring out the common term We continue factoring from where Louise left off: 2 [ 4 x 2 ( x + 3 ) + 5 ( x + 3 )] . We can factor out the common term ( x + 3 ) from the expression inside the brackets.
Completely Factored Form Factoring out ( x + 3 ) , we get: 2 [( 4 x 2 + 5 ) ( x + 3 )] . So the completely factored form is 2 ( 4 x 2 + 5 ) ( x + 3 ) .
Final Answer Comparing our result with the given options, we see that the completely factored form of the polynomial is 2 ( 4 x 2 + 5 ) ( x + 3 ) .
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, ensuring stability and efficiency. Similarly, economists use factoring to analyze economic models and predict market trends. By understanding how to factor polynomials, you can solve a variety of problems in science, engineering, and economics.
The completely factored form of the polynomial 8 x 3 + 24 x 2 + 10 x + 30 is 2 ( 4 x 2 + 5 ) ( x + 3 ) . Therefore, the correct answer from the options provided is C. 2 ( 4 x 2 + 5 ) ( x + 3 ) . This result was obtained by factoring out common terms and grouping the polynomial properly.
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