PLEASE ANSWER ASAP!

Question

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)

Answer 1

You can only cut down a integer number of trees. So you might look at a few integer values for x. As x get large the –x4 term dominates the expression for big losses.

 

x = 0 is easy P(x) = -6. Without cutting any trees you have lost money

 

Put x = 1 and you get for the terms in order  -1 + 1 + 7 -1 -6 = 0. So P(x) crosses zero just before you cut the first tree. So you make a profit on only 1 tree. However when x=10 you are back into no profit.

 

So compute a few values for x = 1,2,3,4,5,6,7,8,9.

 

Someone else surely has a better answer!

Answer 2

We can factor P(x) using division.

 

P(x) = (x - 3)(-x³ - 2x² + x + 2)

 

 

Using division once more,

 

P(x) = (x - 3)(x + 2)(-x² + 1)

 

 

Set f(x) equal to zero.

 

0 = (x - 3)(x + 2)(-x² + 1)

 

 

Set the factors equal to zero.

 

 

x - 3 = 0           x + 2 = 0            -x² + 1 = 0

 

x = 3                x = -2                 -x² = -1

                                                   x² = 1

 

                                                   x = -1        and       x = 1

 

 

Now that we have all the info we need, lets describe the graph.   This is polynomial with degree 4.  Since the coefficient of the leading term is negative and the degree is positive, P(x) starts off to be increasing, and will end decreasing.  In addition, the multiplicity of each factor is odd, which means the roots do not repeat, so P(x) intercepts the x-axis four times.  The first two factors act like a linear function with positive slopes.  The other factor acts like a negative parabola.

The y-intercept is (0, -6).

 

Using this information as best I have described, can the CEO graph P(x) by hand?