# Roots of a polynomial.

• Nov 6th 2008, 12:18 PM
Showcase_22
Roots of a polynomial.
This is a bizarre question I encountered on a homework sheet:

Quote:

Every irreducible polynomial in $\Re$[x] is of degree 1 or 2. Express $x^4+1$ as a product of irreducible polynomials in $\Re$[x]
The only way I can see of doing this is to use complex numbers. I have this idea of writing a root as $"x^2-a"$ where a is a positive real to bypass this. However, I have no way of working out the roots!
• Nov 6th 2008, 12:26 PM
Moo
Hello,
Quote:

Originally Posted by Showcase_22
This is a bizarre question I encountered on a homework sheet:

The only way I can see of doing this is to use complex numbers. I have this idea of writing a root as $"x^2-a"$ where a is a positive real to bypass this. However, I have no way of working out the roots!

Actually, the polynomial can be written as a product this way :
(ax²+bx+c)(dx²+ex+f)

since the leading coefficient is 1, you can take a=1 and d=1
• Nov 6th 2008, 12:49 PM
Laurent
Quote:

Originally Posted by Showcase_22
This is a bizarre question I encountered on a homework sheet:

The only way I can see of doing this is to use complex numbers. I have this idea of writing a root as $"x^2-a"$ where a is a positive real to bypass this. However, I have no way of working out the roots!

One possible trick is to write $X^4+1=(X^4+2X^2+1)-2X^2 = (X^2+1)^2 - (\sqrt{2}X)^2=(X^2+X\sqrt{2}+1)(X^2-X\sqrt{2}+1)$.