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).

Question

janu16 · Answer

@zepdrix

zepdrix · Answer

$$\large\rm P(x)=-x^4+x^3+7x^2-x-6$$

zepdrix · Answer

Do you remember the basic shape of your polynomials?
The leading term (largest degree of x) tells us the `degree of polynomial`.

If the degree is `even` (which it is in this case),
then the "legs" of our function will go off in the same direction,
like in the graph of x^2.

For `odd` degree,
the "legs" go off in opposite directions like in the graph of x^3.

zepdrix · Answer

By "legs" I mean, the end behavior of the function.
So we expect the function to go UP at both ends,
or DOWN at both ends.

Which do you think it will be?
Our leading term is `negative`.
That should give a hint.

janu16 · Answer

yea

janu16 · Answer

down i think

zepdrix · Answer

|dw:1453667478357:dw|Ok good.
We don't know exactly what the function looks like just yet,
but we know what's happening on the ends of it.

janu16 · Answer

yess

zepdrix · Answer

$$\large\rm P(x)=-x^4+x^3+7x^2-x-6$$They want us to find the break-even point, when the profit function is 0,$$\large\rm 0=-x^4+x^3+7x^2-x-6$$This means we're looking for the x values that make this true.
These values go by many names: x-intercepts, roots, zeroes, etc..

zepdrix · Answer

So umm, hmm
maybe we can factor this thing :d

janu16 · Answer

yes so weset set p(x0 to 0 and find x by factoirng

zepdrix · Answer

Hmm it's a 4th degree,
I'm not sure how easy it would be to factor it.

We might want to instead,
find a nice simple root to start with,
and then apply long division.

zepdrix · Answer

have you done `polynomial long division` or `synthetic division`?

janu16 · Answer

ya i have done synthetic division

zepdrix · Answer

lol you were right the first time on the spelling XD

janu16 · Answer

oh lol

zepdrix · Answer

To do division,
we'll first need something that we can divide out.
Plug x=1 or x=-1 into the function.

See if either of those works out to be true.

janu16 · Answer

what do you suggest? x=1? or x= -1?

zepdrix · Answer

x=1 is always a good place to start :)
nice and easy.

if it doesn't work, proceed to x=-1,
if that doesn't work, we can try something else.

janu16 · Answer

i tink 1 will work so we divide eveyrthing by 1?