How does x^4+x^2+1 = (x^2+x+1)(x^2+x+1)?  TOPIC_SOLVED

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How does x^4+x^2+1 = (x^2+x+1)(x^2+x+1)?

Postby gradam on Thu Aug 29, 2013 3:36 pm

Hi!

I'm trying to solve that how does x^4+x^2+1 become (x^2+x+1)(x^2+x+1).
Can someone help me?

I know the goal, but cant figure out how to get there :confused:
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Postby stapel_eliz on Thu Aug 29, 2013 6:45 pm

gradam wrote:I'm trying to solve that how does x^4+x^2+1 become (x^2+x+1)(x^2+x+1).

Are you saying that you need to figure out how to do the factorization, or that you need to confirm that the factorization is correct, or something else? Thank you! :wink:
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Re: How does x^4+x^2+1 = (x^2+x+1)(x^2+x+1)?

Postby gradam on Thu Aug 29, 2013 8:02 pm

Thank you for the fast reply :wink:

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?
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Re: How does x^4+x^2+1 = (x^2+x+1)(x^2+x+1)?  TOPIC_SOLVED

Postby maggiemagnet on Thu Aug 29, 2013 10:31 pm

gradam wrote:...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!!
:clap:
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