# Factoring (X^4+1)

1. Jan 22, 2010

### flyers

1. The problem statement, all variables and given/known data

I am trying to factor x4+1 in to two multiplied polynomials

2. Relevant equations

My teacher gave us this hint that its factored form is (ax2+bx+c)(ax2+bx+c)

3. The attempt at a solution

First i assumed that a and c were equal to 1 so that when x2 is multiplied by the other x2 is gives me x4 and 1 times 1 gives me 1. I knew that b had to be a constant so I multiplied...

(however i didnt know if both b's were the same so i split them into a and b. I also knew one of the constants must be negative so that variables cancel out.)

(x2+ax+1)(x2-bx+1)= x4+1

and I get

x4+ax3-bx3+2x2-abx2+ax-bx+1= x4+1 canceling terms I get

ax3-bx3+2x2-abx2+ax-bx=0

I noticed that to cancel out ax3-bx3 and ax-bx , a and b must be equal to each other. This means 2x2-abx2 = 0

2=ab(but they are the same) 2=b2

b=$$\sqrt{}2$$

So i checked my answer and it works out, but I am wondering if there is a more systematic approach to solve this so that I don't have to assume as much as I did.

2. Jan 22, 2010

### rock.freak667

x4+1 has no real roots, so at most you can factor it into complex roots

using i2=-1

x4-i2=(x2)2-(i)2

3. Jan 22, 2010

### tiny-tim

Welcome to PF!

Hi flyers! Welcome to PF!

(have a square-root: √ )
You could have looked for a way to write it as the difference of two squares …

so complete the square …

x4 + 1 = (x2 + 1)2 - 2x2

4. Jan 22, 2010

### tiny-tim

More generally …

x4 + 2(a-b)x2 + a2

= (x2 + a)2 - 2bx2

= (x2 + (√2b)x + a)(x2 - (√2b)x + a)