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)=-x^4+x^3+7x^2-x-6. Describe to the CEO what the graph looks like. Use complete sentences & focus on the end behaviors of the graph and where the company will break even (where P(x)=o).

Answer 1

i accidentally closed it

Answer 2

Are you allowed to use a graphing calculator?

Answer 3

no i just have to describe what the graph will look like and how i know

Answer 4

But for the second part they ask you to find where P(x) = 0. This is a fourth degree polynomial that will have 4 roots. Without a graphing calculator if we have to find the roots, have you been taught rational roots theorem yet?

Answer 5

um no i dont think so

Answer 6

Do you know synthetic division?

Answer 7

a bit

Answer 8

Okay, for the first part,
The degree of the polynomial is four (because the highest degree here is x^4). This is an even degree and the end behavior of the graph x^4 will be similar to the end behavior of the graph x^2. Here there is a negative sign in front of x^4. Therefore, the end behavior of -x^4 will be similar to the end behavior of -x^2. -x^2 is an inverted parabola with both ends down. So the end behavior of -x^4+x^3+7x^2-x-6 will have both ends down.

This is a fourth degree polynomial and therefore it will have 4 roots. If all those 4 roots are real roots then the graph will cross the x-axis 4 times. If only 2 roots are real then the graph will cross the x-axis 2 times. If no real roots then the graph will not cross the x axis.

Let us find the roots of -x^4+x^3+7x^2-x-6.
Since graphing calculator is not allowed, we have to do solve this by trial and error.
Let us try some small values of x such as x = +1, -1, +2, -2, +3, -3, etc. If any of that x makes the function zero, then we would have found a root. We can do synthetic division and find the quotient and then once again try, small values on the quotient to see what makes it zero and continue in the same manner until all roots are found.

-x^4+x^3+7x^2-x-6
Let us start with x = 1
-(1)^4 + (1)^3 + 7(1)^2 - 1 - 6
-1 + 1 + 7 - 1 - 6 = 0
x = 1 makes the function 0. therefore, (x-1) is a factor.

-x^4+x^3+7x^2-x-6 = (x-1) * { some lower degree polynomial }
To find the lower degree polynomial, divide both sides by (x-1)
Do synthetic division of (-x^4+x^3+7x^2-x-6) / (x-1)

Once you find the quotient with synthetic division, try small values of x to see what makes the quotient 0. Then you would have found another factor. Repeat the process until all roots are found.