Question

How many zeros does this polynomial have: f(x)=2x^3-6x^2+x^5-12x+2

Answer 1

We can use Decartes Rule of Signs

Answer 2

Lets look at the polynomial as it is: this is the positive case:

we count the sign changes: 4 sign changes meaing we have 4 positve zeros

Answer 3

well we actually have 4 postive roots

Answer 4

f(-x)=- - -+
So,we have 1 possible negative root

Answer 5

Indeed there are 5 roots (for a 5th degree polynomial) as suggested by the fundamental theorem of algebra.

However, in this case, we have 1 negative root, 2 positive and two complex roots. So if you plot the function on a calculator, you will only see three roots.

Descartes rule of changes of sign indicates that there is a maximum of four positive roots, but if some roots are complex, they could be reduced, as is the present case.

The roots as indicated on a graph plot shows that the three roots are between -2 and +3, so they are quite easy to pick up.