Prove that 111x^111+11x^11+x+1 has only one real root

1. Jun 25, 2011

1. The problem statement, all variables and given/known data

Prove that the function 111x^111+11x^11+x+1 has exactly one real root.

3. The attempt at a solution

Well, I know how to prove this. f is continuous because It's a polynomial. f(-1) equals -111-11-1+1 = -122 < 0 and f(0) equals 1. therefore, It satisfies all conditions of the intermediate value theorem and the theorem tells us that there must be a zero of f between -1 and 0. Now we can use Rolle's theorem to rule out any other possible real roots of the function. but the problem is the derivative of the function leads to an equation which can't be solved without a computer. so I can't show that the assumption that f has other real roots leads to a contradiction using Rolle's theorem. What can I do now?

2. Jun 25, 2011

micromass

Staff Emeritus

There is no need to calculate the possible roots, you just need to check that there is no root.

As derivative, you've probably found

$$12321x^{110}+121x^{10}+1$$

It suffices to show that there is no positive root (if there is a positive root a, then -a will be a negative root). Now, can you show that the derivative is always >0 for all positive numbers?

3. Jun 25, 2011

How do you say that if a is a positive root then -a will be a negative root? the function is not even. I didn't understand that part.
well, I think I can show that this equation is always positive. what a stupid I am. for any real x: x^10 >= 0. if I multiply it by 121 I'll get 121x^10>=0 Now if I add 1 to both sides I'll get 121x^10+1>=1.
I can do the same for x^110.for any real x: x^110>=0. by summing the two equations I'll get f'(x) >= 1. which proves that It can't have a real root. so, the contradiction!

Thank you micromass xP You are always helpful xP

4. Jun 25, 2011

micromass

Staff Emeritus
Well, it seems you figured it out without using evenness. But the function is even:

$$12321(-x)^{110}+121(-x)^{10}+1=12321x^{110}+121x^{10}+1$$

That's good!

5. Jun 25, 2011