Question

A model predicts the profit, p of a company as p(x) = x3 – 3x2 – 16x + 48, where p is in thousands of dollars and x is number of years passed since January 1, 2011. For which time interval would the company make a loss?

Answer 1

The first step is to determine the zeros of p(x).
From the Remainder Theorem,
p(a) = 0 => (x-a) is a factor of p(x), and x=a is a zero of p(x).

Try x=3:
p(3) = 3^3 - 3*3^2 - 16*3 + 48 = 27 - 27 - 48 + 48 = 0
Therefore x=3 is a zero, and (x-3) is a factor of p(x).

Perform long division.
x² - 16
-------------------------------------
x-3 | x³ - 3x² - 16x + 48
x³ - 3x²
-----------------------------------
- 16x + 48
- 16x + 48

Note that x² - 6 = (x+4)(x-4).

Therefore the complete factorization of p(x) is
p(x) = (x-3)(x+4)(x-4)

To determine when p(x) is negative, we shall test between the zeros of p(x)
x p(x) Sign
---- --------- ---------
-4 0
0 48 +
3 0
3.5 -1.875 -
4 0

p(x) is negative in the interval x = (3, 4).

Answer
The time interval is Jan. 1, 2014 to Jan. 1, 2015.