Which of the following represents the factorization of the trinomial below?
[tex]$-3 x^3-18 x^2-24 x$[/tex]
A. [tex]$-3 x(x-2)(x-4)$[/tex]
B. [tex]$-3 x(x+2)(x+4)$[/tex]
C. [tex]$-3(x^2+2)(x+4)$[/tex]
D. [tex]$-3(x^2-2)(x-4)$[/tex]
Answer (1)
Factor out the greatest common factor: − 3 x 3 − 18 x 2 − 24 x = − 3 x ( x 2 + 6 x + 8 ) .
Factor the quadratic expression: x 2 + 6 x + 8 = ( x + 2 ) ( x + 4 ) .
Combine the factors: − 3 x ( x + 2 ) ( x + 4 ) .
The correct factorization is − 3 x ( x + 2 ) ( x + 4 ) .
Explanation
Understanding the Problem We are given the trinomial − 3 x 3 − 18 x 2 − 24 x and asked to find its factorization from the given options.
Factoring out the GCF First, we factor out the greatest common factor (GCF) from the trinomial. The GCF of − 3 x 3 , − 18 x 2 , and − 24 x is − 3 x . Factoring out − 3 x from the trinomial, we get: − 3 x 3 − 18 x 2 − 24 x = − 3 x ( x 2 + 6 x + 8 ) Now, we need to factor the quadratic expression x 2 + 6 x + 8 .
Factoring the Quadratic We are looking for two numbers that multiply to 8 and add up to 6. The numbers are 2 and 4, since 2 × 4 = 8 and 2 + 4 = 6 . Therefore, the factored form of the quadratic is ( x + 2 ) ( x + 4 ) .
Complete Factorization Therefore, the complete factorization of the trinomial is − 3 x ( x + 2 ) ( x + 4 ) Comparing this result with the given options, we see that option B matches our factorization.
Examples
Factoring trinomials is a fundamental skill in algebra and is used in various real-world applications. For example, if you are designing a rectangular garden and know the area can be represented by the trinomial − 3 x 3 − 18 x 2 − 24 x , factoring it into − 3 x ( x + 2 ) ( x + 4 ) helps you determine possible dimensions for the garden in terms of x . This allows you to plan the layout efficiently based on the available space and resources.
