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anonymous

  • one year ago

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).

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  1. anonymous
    • one year ago
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    @peachpi

  2. anonymous
    • one year ago
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    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

  3. anonymous
    • one year ago
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    alright

  4. anonymous
    • one year ago
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    what do you think this will look like on the right and left ends?

  5. anonymous
    • one year ago
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    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.

  6. anonymous
    • one year ago
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    would the leading coefficient be x4? @peachpi

  7. anonymous
    • one year ago
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    and i'm not sure what the graph would look like @peachpi

  8. anonymous
    • one year ago
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    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.

  9. anonymous
    • one year ago
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    That means you have an EVEN DEGREE polynomial with a POSITIVE LEADING COEFFICIENT.

  10. anonymous
    • one year ago
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    make sense?

  11. anonymous
    • one year ago
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    ohhh that does make sense @peachpi

  12. anonymous
    • one year ago
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    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

  13. anonymous
    • one year ago
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    i think the left and the right would be up

  14. anonymous
    • one year ago
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    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.

  15. anonymous
    • one year ago
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    Now we can solve the equation to find the break-even point. \[x^4-3x^3-8x^2+12x+16=0\]

  16. anonymous
    • one year ago
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    brb

  17. anonymous
    • one year ago
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    alright

  18. anonymous
    • one year ago
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    alright. do you know synthetic division?

  19. anonymous
    • one year ago
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    yeah kinda

  20. anonymous
    • one year ago
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    im not the best at it though

  21. anonymous
    • one year ago
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    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|

  22. anonymous
    • one year ago
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    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 \[(x-4)(x+1)(x+2)(x-2)=0\]

  23. anonymous
    • one year ago
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    with me?

  24. anonymous
    • one year ago
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    yeah sorry im writing some of this down

  25. anonymous
    • one year ago
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    no problem. let me know when you're ready

  26. anonymous
    • one year ago
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    ok im ready now @peachpi

  27. anonymous
    • one year ago
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    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|

  28. anonymous
    • one year ago
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    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|

  29. anonymous
    • one year ago
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    ohhh wow i actually understand this now :D

  30. anonymous
    • one year ago
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    :) cool. Now we just need to make a reasonable guess at the middle.

  31. anonymous
    • one year ago
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    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|

  32. anonymous
    • one year ago
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    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

  33. anonymous
    • one year ago
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    alright

  34. anonymous
    • one year ago
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    The only other thing I think you can do is say the y-intercept is 16|dw:1441293360697:dw|

  35. anonymous
    • one year ago
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    what would the end behaviors be ?

  36. anonymous
    • one year ago
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    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.

  37. anonymous
    • one year ago
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    thank you so much i seriously appreciate all you've helped me with

  38. anonymous
    • one year ago
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    @peachpi

  39. anonymous
    • one year ago
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    you're welcome

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