Factoring X4+1: Step-by-Step Guide

[flyers]

Homework Statement

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

Homework Equations

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

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.

 

[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

 

[tiny-tim]

Welcome to PF!

Hi flyers! Welcome to PF!

(have a square-root: √ )

… 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.

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

 

[tiny-tim]

More generally …

x⁴ + 2(a-b)x² + a²

= (x² + a)² - 2bx²

= (x² + (√2b)x + a)(x² - (√2b)x + a) ​