anonymous
  • anonymous
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).
Algebra
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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anonymous
  • anonymous
@peachpi
anonymous
  • anonymous
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
  • anonymous
alright

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anonymous
  • anonymous
what do you think this will look like on the right and left ends?
anonymous
  • anonymous
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
  • anonymous
would the leading coefficient be x4? @peachpi
anonymous
  • anonymous
and i'm not sure what the graph would look like @peachpi
anonymous
  • anonymous
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
  • anonymous
That means you have an EVEN DEGREE polynomial with a POSITIVE LEADING COEFFICIENT.
anonymous
  • anonymous
make sense?
anonymous
  • anonymous
ohhh that does make sense @peachpi
anonymous
  • anonymous
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
  • anonymous
i think the left and the right would be up
anonymous
  • anonymous
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
  • anonymous
Now we can solve the equation to find the break-even point. \[x^4-3x^3-8x^2+12x+16=0\]
anonymous
  • anonymous
brb
anonymous
  • anonymous
alright
anonymous
  • anonymous
alright. do you know synthetic division?
anonymous
  • anonymous
yeah kinda
anonymous
  • anonymous
im not the best at it though
anonymous
  • anonymous
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|
anonymous
  • anonymous
So now it's \[(x-4)(x^3+x^2-4x-4)=0\] The second part of that can be factored by grouping \[(x-4)[x^2(x+1)-4(x+1)]=0\] \[(x-4)(x+1)(x^2-4)=0\] Then factor the difference of squares \[(x-4)(x+1)(x+2)(x-2)=0\]
anonymous
  • anonymous
with me?
anonymous
  • anonymous
yeah sorry im writing some of this down
anonymous
  • anonymous
no problem. let me know when you're ready
anonymous
  • anonymous
ok im ready now @peachpi
anonymous
  • anonymous
ok so we know the zeros are 4, -1, -2, and 2. Plot them on the x-axis. (please pardon my sorry excuse for points)|dw:1441292680506:dw|
anonymous
  • anonymous
This is where the end behavior comes in. We know that the graph goes up on both ends, so we can do this|dw:1441292840024:dw|
anonymous
  • anonymous
ohhh wow i actually understand this now :D
anonymous
  • anonymous
:) cool. Now we just need to make a reasonable guess at the middle.
anonymous
  • anonymous
basically, (anytime you don't have repeated roots), a polynomial will switch sides of the x-axis when it passes through a zero. So since it's positive to the left of -2, it will be negative between -2 and -1. Then positive again, then negative, then finally positive where we have the right end drawn |dw:1441293088769:dw|
anonymous
  • anonymous
It's really hard to draw a smooth curve on this, so plot it on https://www.desmos.com/calculator to see what it really looks like
anonymous
  • anonymous
alright
anonymous
  • anonymous
The only other thing I think you can do is say the y-intercept is 16|dw:1441293360697:dw|
anonymous
  • anonymous
what would the end behaviors be ?
anonymous
  • anonymous
The end behavior is both ends of the graph go up because the degree is even and the leading coefficient is positive. The y-intercept is 16 so the profit was $16 when 0 laptops were produced. Then profit increases to a local maximum between 0 and 2 produced. Then the company loses money between 2 and 4 produced before breaking even permanently at 4 produced.
anonymous
  • anonymous
thank you so much i seriously appreciate all you've helped me with
anonymous
  • anonymous
@peachpi
anonymous
  • anonymous
you're welcome

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