Use the grouping method to factor this polynomial completely.
[tex]2 x^3+6 x^2+5 x+15[/tex]
A. [tex](2 x^2+3)(x+5)[/tex]
B. [tex](2 x^2+3)(x+3)[/tex]
C. [tex](2 x^2+5)(x+5)[/tex]
D. [tex](2 x^2+5)(x+3)[/tex]
Answer (1)
Group the terms: ( 2 x 3 + 6 x 2 ) + ( 5 x + 15 ) .
Factor out the GCF from each group: 2 x 2 ( x + 3 ) + 5 ( x + 3 ) .
Factor out the common binomial factor: ( 2 x 2 + 5 ) ( x + 3 ) .
The completely factored polynomial is ( 2 x 2 + 5 ) ( x + 3 ) .
Explanation
Understanding the Problem We are given the polynomial 2 x 3 + 6 x 2 + 5 x + 15 and asked to factor it completely using the grouping method.
Grouping Terms First, group the first two terms and the last two terms: ( 2 x 3 + 6 x 2 ) + ( 5 x + 15 )
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, 5 can be factored out: 2 x 2 ( x + 3 ) + 5 ( x + 3 )
Factoring out Common Binomial Now, we can see that ( x + 3 ) is a common binomial factor. Factor it out: ( 2 x 2 + 5 ) ( x + 3 )
Final Answer The factored form of the polynomial is ( 2 x 2 + 5 ) ( x + 3 ) . Comparing this with the given options, we see that option D matches our result.
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. Imagine you're designing a rectangular garden and you know the area can be represented by the polynomial 2 x 3 + 6 x 2 + 5 x + 15 . By factoring this polynomial into ( 2 x 2 + 5 ) ( x + 3 ) , you can determine possible dimensions for the garden, which helps in planning the layout and optimizing the use of space.
