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Polynomial Problem f(x^2+2)=x^4+10x^2+4...

  1. Aug 21, 2015 #1
    so i transferred to a new school and i'm collaborating with another pre-cal teacher. she is kind of helpful but i can tell she doesn't really want to share her work (even though collaboration is about helping each other out).

    She already made a unit test, but i had to make my own answer key. I got stuck with the final problem


    Let f be a polynomial function such that, for all real x,
    f(x^2+2)=x^4+10x^2+4. For all real x, what is f(x^2-2)?

    The only thing I can think of is

    f(x^2+2)= f(x^2)+f(2)

    sadly this is all i can come up with. Any website or youtube link would be grateful to help me in the right direction. Thx
     
  2. jcsd
  3. Aug 21, 2015 #2

    PeroK

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    Your idea is clearly not correct. Polynomials are not linear, so don't work like that.

    Instead, can you see what sort of polynomial ##f## must be?
     
  4. Aug 21, 2015 #3
    An even function with a base of quadratic? Im not sure how to approch it whatsoever, that's why I'm asking for help.
     
  5. Aug 21, 2015 #4

    PeroK

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    Does that mean you think ##f## might be a quadratic?
     
  6. Aug 21, 2015 #5

    ehild

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    You have a function f(g(x)) = x4+10x2+4, with g(x)=x2+2. (Think of composition of functions) What can be the form of the function f(g)?
     
  7. Aug 21, 2015 #6

    HallsofIvy

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    You are given that [itex]f(x^2+ 2)= x^4+ 10x+ 4[/itex]. Let [itex]y= x^2+ 2[/itex] so that [itex]x^2= y- 2[/itex] and [itex]x^4= (y- 2)^2[/itex].

    Now, what is f(y)? What do you get when you replace y with x- 2?
     
  8. Aug 21, 2015 #7

    ehild

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    You meant replacing y by x2-2 :oldsmile:
     
  9. Aug 21, 2015 #8

    HallsofIvy

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    Yes, thank you.
     
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