Consider the equation $\,\,x^{2007}-1+x^{-2007}=0.\,$

Question

I am stuck with the following problem:

Consider the equation $\,\,x^{2007}-1+x^{-2007}=0.\,$Let $\,m$ be the number of distinct complex non-real roots and $\,n$ be the number of distinct real roots of the above equation. Then $\,m-n\,$ is

1.$\,0$

2.$\,2006$

3.$\,2007$

4.$\,4014$

Can someone explain? Thanks in advance for your time.

My Attempt: $y=x^{2007}$ gives $y^2-y+1=0 \implies y=\frac12 \pm \frac{\sqrt 3i}{2}.\,\,$Now, I am not sure how to progress.

Hint: Consider the substitution $y=x^{2007}$. Do you still get stuck? If so, what steps have you tried? — user7530, Oct 31 '13 at 17:56
They are asking how many solutions are there in the equation that are complex but not real. — Carlos Eugenio Thompson Pinzón, Oct 31 '13 at 17:59

score 7 · Accepted Answer · answered Oct 31 '13 at 17:58

7

HINT:

Putting $x^{2007}=a$ we get $a-1+\frac1a=0\iff a^2-a+1=0$

Observe that the discriminant of the last quadratic equation $<0$

Hence the values of $a$ are complex

As $x^{2007}(=a)$ is complex, so will be all the values of $x$

for if $x$ is real, so will be $x^n$ for any integer $n$

Do you know the number of roots of $$x^{2007}-1+x^{-2007}=0$$

answered Oct 31 '13 at 17:58

lab bhattacharjee

274,582

1

Yes .The total number of roots is $4014.$ as $$x^{2007}-1+x^{-2007}=0 \implies x^{4014}-x^{2007}+1=0.$$ Am I right? – learner Oct 31 '13 at 18:08
@learner, so you are done – lab bhattacharjee Oct 31 '13 at 18:09
@learner, as both the values of $a$ are complex, each value of $x$ corresponding to each complex $a$ must be complex, so there will be no real root – lab bhattacharjee Oct 31 '13 at 18:15
so,$m-n=4014-0=4014$ and I am done. – learner Oct 31 '13 at 18:17

score 1 · Answer 2 · edited Jun 12 '20 at 10:38

1

enter image description here

let $A=x^{2007}\,\,\,\,$ new equation is $\,A^2+1=A $ that has no real root

edited Jun 12 '20 at 10:38

Community

1

answered Oct 31 '13 at 18:01

Khosrotash

24,922

3

Seeing your answers so far, why wouldn't you take a look on this particular topic to familiarize yourself with the mathematical formatting on this site. The fact of using an image done inside paint for an equation may result un-clear and time consuming (with all respect, don't get me wrong :-). – TheVal Oct 31 '13 at 18:04

Consider the equation $\,\,x^{2007}-1+x^{-2007}=0.\,$

2 Answers2