Show that the equation x^5+10x+3=0 has exactly one real root. ?

Question

Show that the equation x^5+10x+3=0 has exactly one real root. Rolle's Theorem says that for a continously differentiable function, there has to be a local min or max between any two roots. (a root of the function f is a value a such that f(a)=0).

Answer 1

The function has image negative when x = -1 and has image positive as x=0 .... then there is only a root in the interval [-1,0]



To solve use Numerical Calculus... the root is x = -0,29975 OK!

Answer 2

Use DEsCartes Law of signs

f(x) has no sign changes and hence no positive real roots

f(-x) has one sign change and hence only one real negative root

Thus polynomial has only one real root and it will be negative.

Answer 3

I would graph it