Precalculus: Writing polynomial functions (help!)

  1. Precalculus: Writing polynomial functions (help!) :(

    Hey guys and gals, i'd appreciate if you could help me (well) with this question.

    Write a polynomial function at minimum degree in standard form with real coefficients whose zeros include -1 and 1+4i

    :O its hard for me, im not a math person, and my precalculus teacher doesn't teach, he just talks.

    Thanks!

    btw i have a test tomorrow, so if you could tell me the answer and how you got it, that'd be awesome, because then i can study it and learn, thanks!
     
  2. jcsd
  3. quasar987

    quasar987 4,770
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    First, can you answer the question "what is the minimum degree that a polynomial can have if it has 2 roots" ?
     

  4. No i cannot. What is it?
     
  5. quasar987

    quasar987 4,770
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    Well it's 2. It's can't be one, because a polynomial of degree 1 is of the form

    x+b=0 ==> x=-b is the only root.

    It can be two though because a polynomial of degree 2 with roots a,b can be writen as (x-a)(x-b)=0. But if you set a=-11, b=1+4i, and expand the multiplication of the parenthesis, you'll see that the coefficients are not real. So maybe the minimum degree is 3, in which case the polynomial will be of the form (x+1)(x-(1+4i))(x-z)=0 where z is a 3rd root. Can you choose z such that all the coefficients are real? I've done it and I can't seem to find one, so perhaps it is of degree 4.
     
  6. quasar987

    quasar987 4,770
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    I'm sorry, I'm not much help :p
     
  7. Yeah i know, :P

    Do you think you can simply tell me step-by-step how to do the problem? I dont really understand much of the pre-cal jargon, at all. Im a simple person, :( unfortunately, so if you can, tell me how to do it step-by-step, plz.

    thanx.
     
  8. quasar987

    quasar987 4,770
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    I don't know how!
     
  9. quasar987

    quasar987 4,770
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    I told you about my idea, but it seems impracticable. There must be a simpler way.
     
  10. its alright, i'll ask someone else.

    Thanks for trying though, :D
     
  11. HallsofIvy

    HallsofIvy 40,307
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    You know how to solve polynomial equations, don't you? The simplest way is to factor them.

    If a polynomial has a as a zero, then it has x-a as a factor.

    Also, if a polynomial, with real coefficients, has a+ bi as a zero, it also has a- bi as a factor.

    If a polynomial, with real coefficients, has 1+ 4i as a zero then it must also have 1- 4i as a zero. So your polynomial has x+ 1, x- 1- 4i, and x- 1+ 4i as factors. Multiply them.
     
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