You are having a meeting with the CEO of a technology company. You have interpreted the number of laptops produced versus profit as the function P(x) = x4 -3x3 -8x2 + 12x + 16. Describe to the CEO what the graph looks like. Use complete sentences, and focus on the end behaviors of the graph and where the company will break even (where P(x) = 0).

Question

anonymous · Answer

@peachpi

anonymous · Answer

Let's figure out the end behavior first. Look at the drawing I put up. Which scenario matches your polynomial? http://openstudy.com/users/peachpi#/updates/55e729cee4b0819646d8891a

anonymous · Answer

alright

anonymous · Answer

what do you think this will look like on the right and left ends?

anonymous · Answer

look at the leading coefficient and degree of your function. You need to look at whether the degree is odd or even and whether the leading coefficient is positive or negative. There are 4 different options for end behavior based on those.

anonymous · Answer

would the leading coefficient be x4? @peachpi

anonymous · Answer

and i'm not sure what the graph would look like @peachpi

anonymous · Answer

sorry. I stepped away for a moment.
The degree is the highest exponent. That's 4. 
The leading coefficient is the number in front of $x^4$. There's no number actually written, so the leading coefficient is 1.

anonymous · Answer

That means you have an EVEN DEGREE polynomial with a POSITIVE LEADING COEFFICIENT.

anonymous · Answer

make sense?

anonymous · Answer

ohhh that does make sense @peachpi

anonymous · Answer

ok, so now that we have that info, what will the ends of your graph look like? If you look at that drawing there are 4 options based on degree and leading coefficient
-both ends up
-both ends down
-left end up, right end down
-left end down, right end up

anonymous · Answer

i think the left and the right would be up

anonymous · Answer

Exactly. So as part of your answer you'd say both ends of the graph go up because the degree is even and the leading coefficient is positive.

anonymous · Answer

Now we can solve the equation to find the break-even point.
$$x^4-3x^3-8x^2+12x+16=0$$

anonymous · Answer

brb

anonymous · Answer

alright

anonymous · Answer

alright. do you know synthetic division?

anonymous · Answer

yeah kinda

anonymous · Answer

im not the best at it though

anonymous · Answer

ok. I was trying to factor, but I don't think that will work. Unfortunately it looks like we have to divide. Basically any real roots have to be factors of 16. So typically you have to try a bunch of them and see which works. I've already tried 4 and it works, so we have this.|dw:1441291472156:dw|