How many real roots does this polynomial have? f (x) = 4x^5 + x^3 + 7x - 2 Options: a) 1 b) 3 c) 4 d) 5

I know total number of roots is 5, but how do we find how many of those are real? Please help, thanks.

1 Answer
Dec 3, 2017

a) 1

Explanation:

Given:

f(x) = 4x^5+x^3+7x-2

Note that the pattern of signs of the coefficients is + + + -. With one change of signs, Descartes' Rule of Signs tells us that f(x) has exactly one positive real zero.

Note that:

f(-x) = -4x^5-x^3-7x-2

The pattern of the signs of the coefficients of f(-x) is - - - -. With no changes of signs, Descartes' Rule of Signs tells us that f(x) has no negative real zeros.

So f(x) has exactly one real zero, which is positive.