Answer to Question #114921 in Calculus for Deepak gupta

Question

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

Answer 1

Given polynomial is $f(x)=2x ^5+x^3+5x+1.$

Since the number of changes in sign = 0, by the Descartes rule of sign,

Number of positive real roots = 0.

Now, substitute -x for x in given function,

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

The sign changes from negative (-5x) to positive (1) only, by the Descartes sign rule,

Number of negative real roots = 1.

Also, the degree of f(x) is 5.

That is, the total number of roots = 5.

Remaining roots = 5 - 1 = 4

Hence, Real root is 1 and imaginary roots are 4.