Question

increase the root of the equation x^4+20x

Answer 1

Answer:

First, plot the polynomail to get an idea of what you are doing. Google does the job. You will see that the polynomial has only one root near −2. Now you must prove it.

The derivative is

f′(x)=20(x4+x3−x−1)=20(x3−1)(x+1)

It has two real roots at −1 and 1 as suggests the graphic of f. Since none of them are multiple roots, the derivative change its sign at them, so f is increasing in (−∞,−1] and in [1,∞), and it is decreasing in [−1,1].

Therefore, f has a minimum at 1: f(1)=19>0. So f has only one root.