P(x) = x^4 + x^3 + x^2 + x + 11

Question

Use Descartes' Rule of Signs to determine how many positive and how many negative real zeros the polynomial can have. Then determine the possible total number of real zeros. (Enter your answers as comma-separated lists.)

number of positive zeros possible
number of negative zeros possible
number of real zeros possible

Answer 1

Thus, there are no sign changes from the first term to the

last and as such, there are no possible positive real roots.

This results in 4 sign changes from the first term to the last

and thus results in the potential of 4 possible negative real roots.

Answer 2

It is obvious that this polynomial can have NO positive real roots because for x≥0 the function is greater than 0 and this is consistent with Descartes' Rule of Signs.

Th same rule says that there should be a maximum of 4 negative real roots (4 sign changes)...but, in fact, there are none.