According to Rolle's Thereom, what is the largest # of real roots that x^7 + x can have?

Answer 1

I don't think Rolle's theorem is very useful here. There are at least 2 almost immediate ways we can see there's just one real root.

1) p(x) = x^7 + x is a strictly increasing function, which goes to -oo when x goes to -oo and to oo when x goes to oo. Therefore, it's a bijection that takes on once and only once every real number. This implies p(x) = 0 happens for one and only one x, namely x = 0.

2) p(x) = x(x^6 +1), so that the the roots of this polynomial are 0, a real number, and the roots of x^6 = -1. But since this last equation has no real roots, if follos x = 0 is the only real root of p.

Answer 2

x^7+x is an increasing function, so it can cross the x axis at most one time. Since it in fact does cross at x=0, it has exactly one real root.