Solving Higher Degree Polynomial For Real Solution(s).

[anemone]

Find real solution(s) to the equation \(\displaystyle (x^2-9x-1)^{10}+99x^{10}=10x^9(x^2-1)\)

 

[Opalg]

Find real solution(s) to the equation \(\displaystyle (x^2-9x-1)^{10}+99x^{10}=10x^9(x^2-1)\)

Hint 1:

Spoiler

Divide both sides by $x^{10}$.

Hint 2:

Spoiler

Let $y = x-\frac1x$.

Hint 3:

Spoiler

What happens when $y=10$?

That will give you two real solutions. There are two others, but I do not think that they can be explicitly determined.

 

[anemone]

Hi Opalg,

I thank you for taking the time to share your helpful hints in this challenge problem.

My approach is different from yours and I wish to share it here too.

Hint 1:

Spoiler

We first rewrite the equation as \(\displaystyle (x^2-9x-1)^{10}-10x^9(x^2-9x-1)+9x^{10}=0\)

Hint 2:

Spoiler

Assume that \(\displaystyle (x^2-9x-1)^{10}+9x^{10}=10x^9(x^2-9x-1)\) is true.

Hint 3:

Spoiler

We need to show that \(\displaystyle (x^2-9x-1)^{10}=x^{10}\) and \(\displaystyle x^9(x^2-9x-1)=x^{10}\) are true by obtaining two equivalent equation, i.e. \(\displaystyle x^2-10x-1=0\) in order to prove our assumption is true.

Hint 4:

Spoiler

The original problem is solved if we solve for x for \(\displaystyle x^2-10x-1=0\)

That will give you two real solutions. There are two others, but I do not think that they can be explicitly determined.

I failed to realize that there are two other real solutions to this problem. Would you mind to teach me how to tell how many real solutions there are for this particular problem?

Thanks in advance.

 

[Opalg]

I failed to realize that there are two other real solutions to this problem. Would you mind to teach me how to tell how many real solutions there are for this particular problem?

That was my silly mistake. There are no other real solutions. Sorry about that.

 

[anemone]

That was my silly mistake. There are no other real solutions. Sorry about that.

I see...hey, don't worry about that! :)

You are and always will be one of my favorite mathematicians at this site!:p:)

 

[topsquark]

Find real solution(s) to the equation \(\displaystyle (x^2-9x-1)^{10}+99x^{10}=10x^9(x^2-1)\)

Graph it! Hey, I'm a Physicist. (Beer) (Physics is always better with beer.)

-Dan

 

[Nono713]

Graph it! Hey, I'm a Physicist. (Beer) (Physics is always better with beer.)

-Dan

Indeed. I am now confident the real roots are between $-\infty$ and $+\infty$, more or less (Smoking)

 

[chisigma]

Graph it! Hey, I'm a Physicist. (Beer) (Physics is always better with beer.)

-Dan

Unfortunately in this case graphing in not a comfortable way to arrive to the solution (Wasntme)...

http://www.123homepage.it/u/i69151835._szw380h285_.jpg.jfif

All is allways better with beer! (Beer)...Kind regards $\chi$ $\sigma$