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

• Dec 8th 2010, 06:47 AM
abrahamtim
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 !
• Dec 8th 2010, 07:44 AM
ark600
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 :) )