Example: 3x^4 -x^3 -15x^2 -x +2

It goes from positive (3x^4) to negative (-x^3) so that’s 1 sign change.

It goes from negative (-x^3) to (-15x^2) so there’s no sign change.

It goes from negative (-15x^2) to negative (-x) so there’s no sign change.

It goes from negative (-x) to positive (2) so that’s 1 sign change.

Add all the sign changes up. 1+0+0+1=2

Now try to subtract 2 from your answer, so that it equals no less than 0. (2-2=0. It’s doable.)

So for the positive zeros, you can have 2 or 0 positive real zeros. (the 2 comes from the original answers, and the 0 comes from our second answer when we subtracted 2.)

To find the negative zeros, change the sign for all terms that have an odd exponent.

Ex: 3x^4 -x^3 -15x^2 -x +2

Only -x^3 and -x (same as -x^1) have an odd exponent. So change their signs to x^3 and x^1. Now rewrite the equation:

3x^4 +x^3 -15x^2 +x +2

Now count the sign changes. The only sign changes are between x^3 and -15x^2, and between -15x^2 and x. So that’s 2 changes. Can you subtract 2 from that answer so that it equals no less than zero? (2-2=0. It works!)

So there are 2 or 0 negative zeros.