...how does x^4+x^2+1 become (x^2+x+1)(x^2+x+1).
This was an exercise from my math workbook about factorization. After a lot of tries I gave up and checked the answer, but I still couldn't figure out how to do the factorization. What's the method of that?
I don't know. This doesn't have any x-intercepts, so you can't find zeroes and then go backwards from that. It's a square, for sure, but I'm not seeing how you're supposed to "see" that in the first place. I did find a trick online, though; maybe you're supposed to "know" this??
x^4 + x^2 + 1
x^4 + 1 + x^2
x^4 + 1 + x^2 + 2x^2 - 2x^2 <== [this is the "trick"!]
x^4 + 2x^2 + 1 + x^2 - 2x^2
(x^2 + 1)^2 - x^2
(x^2 + 1)^2 - (x)^2
Then do difference of squares:
[(x^2 + 1) + (x)][(x^2 + 1) - (x)]
[x^2 + 1 + x][x^2 + 1 - x]
[x^2 + x + 1][x^2 - x + 1]
By the way, this solution shows that the "answer" they gave you in the back is wrong!!