Home > Math Group > Question and Answer

Math Group

Question:

fairydrink
fairydrink
Student
High School - 10th Grade

Find the multiple roots of the polynomial, if there are any:

(n-1)X^n - n*X^(n-1)+1

Rate question:

Posted by fairydrink on Wednesday April 29, 2009 at 6:14 AM and tagged with math, multiple root, polynomial.

Answers:

  1. neela
    neela Teacher
    Graduate School

    eNotes Editor

    Let  f(x) = (n-1)x^n - n*x^(n-1)+1

    If f(x) has  r mutiple roots , then f '(x) has  same r-1 mutiple roots.

    f'(x) = d/dx{(n-1)x^n -nx^(n-1) +1 }

    =n(n-1)x^(n-1) -n(n-1)x^(n-2)

    =n(n-n)x^(n-2){x-1}, Thus x =0 or x-1=0 or  x=1   are  the roots of  f'(x).

    But x= 0 is not a root of f(x) , as f(0) =

    =( n-1)*0-n*0*+1 =1  and not =0.

    When x-1 =0 or x=1 , f(1) =(n-1)*1-(n)*1+1 =0. Therefore, x=1 is a mutiple root of  f(x) order of multiplicity  is  2 i.e one higher than that of f'(x).

     

    Rate answer:

    Posted by neela on Tuesday May 26, 2009 at 1:57 AM