Results 1 to 3 of 3

Math Help - Sum of Roots of a polynomial

  1. #1
    Member
    Joined
    Nov 2009
    Posts
    169

    Sum of Roots of a polynomial

    Let p(z) = a_0 + a_1 z + \ldots + a_d z^d be a complex polynomial with coefficients a_i \in \mathbb{C} of degree d.

    1) Let x_1,\ldots,x_d be the roots of p. Show that \sum_{i=1}^d x_i  = -\frac{a_{d-1}}{a_d}.

    2) Let v \in \mathbb{C} and y_1,\ldots,y_d satisfy p(y_i) = v \forall 1 \leq i \leq d
    What is \sum_{i=1}^d y_i ?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    5
    1) is...

    \displaystyle  p(z)= a_{d}\ z^{d} + a_{d-1}\ z^{d-1} + ... + a_{1}\ z + a_{0} = a_{d}\ \prod_{i=1}^{d} (x-x_{i}) (1)

    ... so that...

    \displaystyle \frac{a_{d-1}}{a_{d}} = -\sum_{i=1}^{d} x_{i} (2)

    Kind regards

    \chi \sigma
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by EinStone View Post
    Let p(z) = a_0 + a_1 z + \ldots + a_d z^d be a complex polynomial with coefficients a_i \in \mathbb{C} of degree d.

    1) Let x_1,\ldots,x_d be the roots of p. Show that \sum_{i=1}^d x_i  = -\frac{a_{d-1}}{a_d}.


    This is almost trivial if you write p(z) = a_0 + a_1 z + \ldots + a_d z^d=a_d(z-x_1)\cdot\ldots\cdot(z-x_d) and compare

    coefficients of respective powers of z.



    2) Let v \in \mathbb{C} and y_1,\ldots,y_d satisfy p(y_i) = v \forall 1 \leq i \leq d
    What is \sum_{i=1}^d y_i ?


    Use (1) with the polynomial g(z) = p(z)-v

    Tonio

    .
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. roots of i, polynomial
    Posted in the Algebra Forum
    Replies: 3
    Last Post: February 28th 2011, 06:08 AM
  2. Polynomial Roots
    Posted in the Algebra Forum
    Replies: 4
    Last Post: September 23rd 2010, 11:20 PM
  3. polynomial roots
    Posted in the Algebra Forum
    Replies: 4
    Last Post: September 1st 2009, 12:30 PM
  4. Roots of a polynomial.
    Posted in the Math Topics Forum
    Replies: 2
    Last Post: November 6th 2008, 11:49 AM
  5. Polynomial Roots
    Posted in the Calculus Forum
    Replies: 8
    Last Post: November 25th 2007, 09:37 AM

Search Tags


/mathhelpforum @mathhelpforum