1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Constructing Polynomials

  1. Aug 22, 2009 #1
    There's a question in Calculus by Spivak about polynomials and I was wondering about how to construct them to have specific roots or values at certain points. For example it says if
    [tex]x_{1}[/tex], ..., [tex]x_{n}[/tex] are distinct numbers, find a polynomial [tex]f_{i}[/tex] such that it's of degree n-1 which is 1 at [tex]x_{i}[/tex] and 0 at [tex]x_{j}[/tex] for [tex]j \neq i[/tex]. Now I know that for roots it's simply the product [tex]\prod (x-x_{j})[/tex] running from j=1 to n and [tex]j \neq i[/tex]. That evaluates the polynomial to 0 correctly. But I don't know how to add the additional condition of [tex]x_{i}[/tex] to make it evaluate to 1. I know I can't multiply another factor [tex](x-x_{k})[/tex] because that would increase the degree to n and it wouldn't do any good anyway. Is it a piecewise defined function? That seems a little too trivial. Any hints you could give me?
     
  2. jcsd
  3. Aug 22, 2009 #2
    hehe i remember doing this question a few months ago

    What value does the polynomial you have at the moment take at x_i?
     
  4. Aug 22, 2009 #3

    HallsofIvy

    User Avatar
    Science Advisor

    Multiply that product by some number, a: [tex]a\prod_{j\ne i} (x-x_{j})[/tex].

    Now you can choose a to make the value at [tex]x= x_i[/tex] 1. That is, you want [tex]a\prod (x_i-x_{j})= 1[/tex] so you must have [tex]a= \frac{1}{\prod (x_i- x_j)}[/tex].

    That gives [tex]\frac{\prod (x- x_j)}{\prod (x_i- x_j)}[/tex]

    In fact, by adding things of that type, you can get a polynomial of degree n-1 that takes on specified values at n points- the "Lagrange Interpolation Polynomial".
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook




Loading...