Question

A mysterious box is delivered to the dinner party you are attending. The label on the box says that the volume of a box is the function f(x) = x3 + 3x2 – 10x – 24. To open the box, you need to identify the correct factors of f(x). Partygoers offer up solutions, and it is your job to find the right ones.
Their suggestions are:
•(x – 1)
•(x + 2)
•(x – 3)
•(x + 4)
•(x + 6)
•(x – 12)


List the correct factors. Then justify your selections with complete sentences.

Answer 1

F(x) = x³ + 3x² - 10x - 24

The root factors must be the factors of 24.
1 x 24 ; 2 x 12 ; 3 x 8 ; 4 x 6.

The most probable roots would be -4, -3, -2, -1, 1, 2, 3, or 4.

Using synthetic division, one can find that the smallest root is -3.
(x - 3)(x² - 6x - 8). The quadratic can now be factored as (x + 2)(x - 4) and further factored as (x - 3)(x + 2)(x - 4). 

The correct factors are: (x - 3) (x + 2) (x - 4)