Question

find the nature of the roots of the polynomial 2x^5+x^3+5x+1

Answer 1

We have to find the nature of the roots of the polynomial $2x^5+x^3+5x+1$.

Assume that $f(x)=2x^5+x^3+5x+1$. (1)

By the Descartes rule of signs, number of positive real real roots is 0 (because the number of changes of signs is 0).

Let's substitute -x to the (1):

$f(-x)=2(-x)^5+(-x)^3+5(-x)+1$

$f(-x)=-2x^5-x^3-5x+1$ (2)

We see that in (2) the sign changes only once (from negative (-5x) to positive (1)).

Thus, by the Descartes rule of signs, number of negative real roots is 1 (because the number of changes of signs is 1).

By the condition, the degree of $f(x)$ is 5. Thus, by the fundamental theorem of algebra, the total number of roots is 5. So, remaining roots $=5-1=4.$ These roots are imaginary.

Answer: 1 real negative and 4 imaginary roots.