# Solving Higher Degree Polynomial For Real Solution(s).

#### anemone

##### MHB POTW Director
Staff member
Find real solution(s) to the equation $$\displaystyle (x^2-9x-1)^{10}+99x^{10}=10x^9(x^2-1)$$

#### Opalg

##### MHB Oldtimer
Staff member
Find real solution(s) to the equation $$\displaystyle (x^2-9x-1)^{10}+99x^{10}=10x^9(x^2-1)$$
Hint 1:
Divide both sides by $x^{10}$.
Hint 2:
Let $y = x-\frac1x$.
Hint 3:
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

##### MHB POTW Director
Staff member
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:
We first rewrite the equation as $$\displaystyle (x^2-9x-1)^{10}-10x^9(x^2-9x-1)+9x^{10}=0$$

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

Hint 3:
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:
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?

#### Opalg

##### MHB Oldtimer
Staff member
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

##### MHB POTW Director
Staff member
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!

#### topsquark

##### Well-known member
MHB Math Helper
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. (Physics is always better with beer.)

-Dan

#### Bacterius

##### Well-known member
MHB Math Helper
Graph it!! Hey, I'm a Physicist. (Physics is always better with beer.)

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

#### chisigma

##### Well-known member
Graph it!! Hey, I'm a Physicist. (Physics is always better with beer.)

-Dan
Unfortunately in this case graphing in not a comfortable way to arrive to the solution ...

All is allways better with beer! ...

Kind regards

$\chi$ $\sigma$