Use Descartes' Rule to determine the possible numbers of positive and negative zeros. 𝑓(𝑥)=𝑥^7−3

Answer 1

f(x) = x⁷ - 3 has degree 7, so there are a total of 7 roots including any repetitions or complex numbers.

f(x) has 1 sign change, so the number of positive real roots is 1.

f(-x) = (-x)⁷ - 3 = -x⁷ - 3. Since there are no changes in sign, there are no negative real roots.

So, there are 6 complex roots, 1 positive real root, and no negative real roots.