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You are having a conference call with the CEO of a paper company. You have interpreted the number of trees cut down versus profit as the function P(x) = –x4 + x3 + 7x2 − x − 6. Describe to the CEO what the graph looks like and, in general, how to sketch the graph without using technology. Use complete sentences, and focus on the end behaviors of the graph and where the company will break even (where P(x) = 0).

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P(x) = –x4 + x3 + 7x2 − x − 6

The graph will look like a bell, or a downward facing parabola, since the dominant term is -x^4

The graph has a y-intercept of -6, which can be considered the fixed cost (cost of zero trees cut) and moves upward from there, eventually reaching a maximum at some positive number of trees and dropping again. As x gets very large, profit becomes a very large negative number. Since we can't have negative trees cut, we're only interested in positive x values

The company breaks even at P(x) = 0
0 =  –x4 + x3 + 7x2 − x − 6, x greater than zero
0 = (x+2)(x+1)(x-1)(x-3)
The company breaks even at 1 tree and 3 trees. 

Answer:below

Step-by-step explanation:

Using synthetic division, the left side factors to

=(x - 3)(-x3 - 2x2 + x + 2) = 0

Then group the second factor,

(x - 3)[-x3 + x - 2x2 + 2] = 0

(x - 3)[-x(x2 - 1) - 2(x2 - 1)] = 0

(x - 3)(x2 - 1)(-x - 2) = 0

(x - 3)(x - 1)(x + 1)(-x - 2) = 0

Our zeros are then

x = -2;x = -1;x = 1;x = 3

- Since we have a negative 4th degree function, the graph will start off increasing, and then end off decreasing

- This is a symmetrical function

-The company makes a maximum profit when 2 trees are cut down.  When no trees are cut down, the company loses profit by 6.

-The profits start to increase when at most 2 trees are cut down, after 2 trees are cut down the profits decrease.

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