Roots of a polynomial.

**Showcase_22** · November 6th 2008, 11:18 AM

This is a bizarre question I encountered on a homework sheet:

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!

**Moo** · November 6th 2008, 11:26 AM

Hello,

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

**Laurent** · November 6th 2008, 11:49 AM

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 $X^4+1=(X^4+2X^2+1)-2X^2 = (X^2+1)^2 - (\sqrt{2}X)^2=(X^2+X\sqrt{2}+1)(X^2-X\sqrt{2}+1)$ .

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