A community for students.
Here's the question you clicked on:
 0 viewing
anonymous
 one year ago
I need some help with understanding a basic mathematical proof.
anonymous
 one year ago
I need some help with understanding a basic mathematical proof.

This Question is Closed

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Disprove the statement: There is a real number x such that \[x ^{6}+x ^{4}+1 = 2x ^{2}\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Proof: For all real numbers, x, \[x ^{6}+x ^{4}+1\neq 2x ^{2}\].

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Case 1: suppose x = 0, then x^6 + x^4 + 1 = 1 and 2x^2 = 0, so x^6 + x^4 +1 =/= 2x^2.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Case 2 (this is the one I'm having a little trouble understanding) : Suppose \[x \neq0\], then x^2 >0. Since \[x ^{6}=x ^{2^{3}}\], x^6 >0. Therefore x^6+(x^21)^2 >x^6, so x^6 +x^4 +1  2x >0

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0(x^21)^2 is factored from x^6+x^4+12x^2...can I end the proof there?

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.1x is nonzero > x^2 is positive x^4, x^6, etc are all positive as well

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.1and as you pointed out, x^42x^2+1 turns into (x^21)^2 so (x^21)^2 is always positive when x is nonzero

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Yes, so x^6+x^4+1  2x^2 >0 ...can I say that x^6+x^4+1 >2x^2? Thus x^6+x^4+1 =/= 2x^2

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.1I would do it like this x^6+x^4+1  2x^2 >0 x^6+x^4 2x^2+1 >0 x^6+(x^2 1)^2 >0 when x is nonzero, x^6 is positive and (x^2 1)^2 is positive so that's why x^6+(x^2 1)^2 >0 is true and x^6+(x^2 1)^2 = 0 will never be true

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0I understand what you did, but how (exactly) does that show x^6+x^4+1 =/= 2x^2?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Or is that supposed to be obvious so I don't have to write anything more?

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.1it shows that x^6+(x^2 1)^2 =/= 0

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.1which consequently means x^6+x^4+1  2x^2 =/= 0

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0ahh, yes so bringing 2x^2 to the left side gives x^6+x^4+1 =/= 2x^2
Ask your own question
Sign UpFind more explanations on OpenStudy
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.