To find the possible number of positive roots, look at the signs on the coefficients and count the number of times the signs on the coefficients change from positive to negative or negative to positive.

Since there are sign changes from the highest order term to the lowest, there are at most positive roots (Descartes’ Rule of Signs). The other possible numbers of positive roots are found by subtracting off pairs of roots .

Positive Roots: or

To find the possible number of negative roots, replace with and repeat the sign comparison.

Apply the product rule to .

Raise to the power of .

Multiply by .

Apply the product rule to .

Raise to the power of .

Multiply by .

Multiply by .

Since there are sign changes from the highest order term to the lowest, there are at most negative roots (Descartes’ Rule of Signs). The other possible numbers of negative roots are found by subtracting off pairs of roots (e.g. ).

Negative Roots: or

The possible number of positive roots is or , and the possible number of negative roots is or .

Positive Roots: or

Negative Roots: or

Determine the Possible Number of Real Roots f(x)=12x^5-4x^3+4x+8