1. Not finding help here? Sign up for a free 30min 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!

Polynomial Help

  1. Oct 2, 2005 #1
    If m, n, and 1 are non-zero roots of the equation [tex]x^3 - mx^2 + nx - 1 = 0[/tex], then find the sum of the roots

    This is what I did..

    m, n, 1 are the roots. m and n not equal to 0

    [tex]x^3 - mx^2 + nx - 1 = 0[/tex]
    f(m) = 0 --> m^3 - m^3 + mn - 1
    1 = mn (1)

    f(1) = 1 - m + n - 1 = 0
    ... m = n (2)

    Sub (2) --> (1)

    [tex]m^2 = 1 [/tex]
    [tex]m = +/- 1

    since m and n are equal... the roots must be either 1, 1, 1 or -1, -1, 1. The answer on the sheet says the answer is -1. So my question is, how do we determine which roots are the answers. Thanks
     
  2. jcsd
  3. Oct 2, 2005 #2

    hotvette

    User Avatar
    Homework Helper

    This might work. If m, n, and 1 are the roots, that means (x-m)*(x-n)*(x-1) = 0. If you multiply this out and substitute the 2 different sets of values for m & n, I'm guessing that only 1 set is valid.
     
    Last edited: Oct 2, 2005
  4. Oct 2, 2005 #3
    oops nvm it didn't work
     
    Last edited: Oct 2, 2005
  5. Oct 8, 2005 #4

    hotvette

    User Avatar
    Homework Helper

    looks like there are 2 possible solutions.
     
  6. Oct 8, 2005 #5

    AKG

    User Avatar
    Science Advisor
    Homework Helper

    There is only one solution. I used long division. If you divide your polynomial by (x-1), you know you should get a quadratic polynomial, but you get:

    x² + (1-m)x + (n+1-m) + (n-m)/(x-1)

    So you know n - m = 0. This is one useful fact. Now you're left with:

    x² + (1-m)x + (n+1-m)

    You know that m is a root of this polynomial, so dividing by (x-m) should give a linear polynomial. When you do the long division, you get:

    x + 1 + (n+1)/(x-m)

    So you know n + 1 = 0. This gives n = -1. Having already got that n - m = 0, you now get m = -1. And you have the other root being 1.

    (-1) + (-1) + 1 = -1.
     
  7. Oct 8, 2005 #6

    hotvette

    User Avatar
    Homework Helper

    Good one AKG! :redface:
     
  8. Oct 9, 2005 #7

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    In your very first post, you note that the solutions must be either {1, 1, 1} or {1, -1, -1} and ask how can you tell that only {1, -1, -1} is correct.

    If {1, 1, 1} were the roots, then the equation could be written
    (x-1)(x-1)(x-1)= 0 but (x-1)(x-1)(x-1)= x3- 3x2+ 3x- 1, not x3- x2+ x- 1 as it should be with m=n= 1.

    If {1, -1, -1}, on the other hand, were the roots then the equation could be written (x-1)(x+1)(x+1)= (x2- 1)(x+1)= x3+x2- x- 1 as required.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Polynomial Help
  1. Polynomial Help (Replies: 11)

  2. Polynomial help (Replies: 10)

  3. Polynomials help (Replies: 5)

Loading...