Show that the polynomial $$P(x):=x^4-6x+6$$ has no real roots.
We need to solve this problem without using calculus. This is a problem from my son's olympiad textbook.
Since the degree of the polynomial is $4$, our task seems difficult. I tried to factor the polynomial. For example, I was expecting an expression like $(x^2+1)(x^2+x+1)$. But unfortunately it didn't.
Rational root theorem obviously fails. Because, real roots don't exist.
I tried to rewrite the polynomial
$$x^4-6x+6=(x^2+ax+b)(x^2+cx+d)$$
So, it sufficies to show that $a^2-4b<0$ and $c^2-4d<0$.
But, I am not sure, is it good track or not.
Thanks for advance .
Best Answer
Write $$x^4-6x+6=(x^2-1)^2+2\left(x-\frac32\right)^2+\frac12$$ and then it is clear that $x^4-6x+6>0$ for all $x\in\mathbb{R}$.