What is the factorization of the trinomial below?
[tex]$-2 x^3+2 x^2+12 x$[/tex]
A. [tex]$2 x(x^2+3)(x+2)$[/tex]
B. [tex]$-2 x(x+3)(x-2)$[/tex]
C. [tex]$-2 x(x+3)(x+2)$[/tex]
D. [tex]$-2 x(x-3)(x+2)$[/tex]
Answer (1)
Factor out the greatest common factor: − 2 x 3 + 2 x 2 + 12 x = − 2 x ( x 2 − x − 6 ) .
Factor the quadratic expression: x 2 − x − 6 = ( x − 3 ) ( x + 2 ) .
Combine the factors: − 2 x ( x − 3 ) ( x + 2 ) .
The correct factorization is: − 2 x ( x − 3 ) ( x + 2 ) .
Explanation
Problem Analysis We are given the trinomial − 2 x 3 + 2 x 2 + 12 x and asked to factor it. We need to identify the correct factorization from the given options.
Factoring out the GCF First, we look for the greatest common factor (GCF) of the terms in the trinomial. The GCF of − 2 x 3 , 2 x 2 , and 12 x is − 2 x . Factoring out − 2 x from the trinomial, we get: − 2 x 3 + 2 x 2 + 12 x = − 2 x ( x 2 − x − 6 )
Factoring the Quadratic Next, we need to factor the quadratic expression x 2 − x − 6 . We are looking for two numbers that multiply to − 6 and add to − 1 . These numbers are − 3 and 2 . Therefore, we can write the quadratic as: x 2 − x − 6 = ( x − 3 ) ( x + 2 )
Complete Factorization Now, we substitute the factored quadratic back into the expression: − 2 x ( x 2 − x − 6 ) = − 2 x ( x − 3 ) ( x + 2 ) So, the complete factorization of the trinomial is − 2 x ( x − 3 ) ( x + 2 ) .
Identifying the Correct Option Comparing our result with the given options, we see that option D matches our factorization: D. − 2 x ( x − 3 ) ( x + 2 )
Examples
Factoring trinomials 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 analyzing systems. Imagine you are designing a bridge and need to calculate the load-bearing capacity. The equation representing the load might be a complex trinomial, and factoring it simplifies the equation, making it easier to find the critical points and ensure the bridge's safety. Similarly, in economics, factoring can help simplify equations related to cost, revenue, and profit, allowing businesses to make informed decisions.
