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Math Help - greatest common divisor of two polynomials f,g ? Is it unique ?

  1. #1
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    greatest common divisor of two polynomials f,g ? Is it unique ?

    a)
    Define the greatest common divisor of two polynomials f,g. Is it unique ?

    b)
    Find the greatest common divisor of the polynomials f :  x^4 + x - 2 and g = x^2 - 1

    c)
    The polynomial  f = x^4 - x^3 -7x^2 + x + 6 has four real zeros, given that 1 and 3 are zeros, find the other two ?

    This is a past paper without solutions, it would be great if you could help me with these to revise for my exam, thanks alot !
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  2. #2
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    a) The gcd is the highest order polynomial that divides ( with no remainder) both f and g. It is unique.

    b) x^4+x-2 = (x^2-1)(x^2+1) + (x-1)
    and (x^2-1)=(x-1)(x+1) so x-1 is gcd

    c) Since 1 and 3 zeroes you know that (x-1)(x-3) divides f.
    So divide f by (x-1)(x-3)=x^2-4x+3 :

    x^4-x^3-7x^2+x+6 = (x^2-4x+3)(x^2+3x+2)

    so the other two zeroes are the zeroes of x^2+3x+2= (x+1)(x+2), namely -1 and -2.

    ( I think this is correct - I forgot how that long division of polynomial is a pain )
    Last edited by mr fantastic; December 25th 2010 at 06:04 PM. Reason: Restored deleted post. User banned.
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