Which of the following represents the factorization of the trinomial below?
[tex]6 x^2+11 x+3[/tex]
A. [tex](3 x+1)(2 x+3)[/tex]
B. [tex](3 x+2)(2 x+3)[/tex]
C. [tex](3 x+3)(2 x+2)[/tex]
D. [tex](3 x+3)(2 x+1)[/tex]
Answer (1)
Expand each of the given options.
Compare the expanded form of each option with the given trinomial 6 x 2 + 11 x + 3 .
The option whose expanded form matches the given trinomial is the correct factorization.
The correct factorization is ( 3 x + 1 ) ( 2 x + 3 ) .
Explanation
Understanding the Problem We are given the trinomial 6 x 2 + 11 x + 3 and four possible factorizations. Our goal is to determine which of the given options is the correct factorization. We can do this by expanding each option and comparing it to the given trinomial.
Expanding the Options Let's expand each of the given options:
A. ( 3 x + 1 ) ( 2 x + 3 ) = 3 x ( 2 x ) + 3 x ( 3 ) + 1 ( 2 x ) + 1 ( 3 ) = 6 x 2 + 9 x + 2 x + 3 = 6 x 2 + 11 x + 3
B. ( 3 x + 2 ) ( 2 x + 3 ) = 3 x ( 2 x ) + 3 x ( 3 ) + 2 ( 2 x ) + 2 ( 3 ) = 6 x 2 + 9 x + 4 x + 6 = 6 x 2 + 13 x + 6
C. ( 3 x + 3 ) ( 2 x + 2 ) = 3 x ( 2 x ) + 3 x ( 2 ) + 3 ( 2 x ) + 3 ( 2 ) = 6 x 2 + 6 x + 6 x + 6 = 6 x 2 + 12 x + 6
D. $(3x+3)(2x+1) = 3x(2x) + 3x(1) + 3(2x) + 3(1) = 6x^2 + 3x + 6x + 3 = 6x^2 + 9x + 3
Comparing with the Trinomial Comparing the expanded forms with the given trinomial 6 x 2 + 11 x + 3 , we see that option A matches exactly.
Final Answer Therefore, the correct factorization of the trinomial 6 x 2 + 11 x + 3 is ( 3 x + 1 ) ( 2 x + 3 ) .
Examples
Factoring trinomials is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use factoring to design structures, ensuring stability and optimal use of materials. Imagine designing a rectangular garden where the area is represented by the trinomial 6 x 2 + 11 x + 3 . By factoring this trinomial into ( 3 x + 1 ) ( 2 x + 3 ) , you determine the dimensions of the garden, allowing you to plan the layout efficiently. This ensures that the garden fits perfectly within the available space and maximizes the planting area.
