73. A classic theorem, Descartes' Rule of Signs, tells us about the number of positive and negative real zeros of a polynomial function, by looking at the polynomial's variations in sign. A variation in sign occurs when consecutive coefficients (in standard form) have opposite signs.
If f(x)=anxn+⋯+a0 is a polynomial of degree n, then
- The number of positive real zeros of f is equal to the number of variations in sign of f(x), or that number less some even number.
- The number of negative real zeros of f is equal to the number of variations in sign of f(−x), or that number less some even number.
Use Descartes' Rule of Signs to determine the possible numbers of positive and negative real zeros of the function.
(a) f(x)=x3+x2−x+1
(b) f(x)=x3+x2+x+1
(c) f(x)=2x3+x−3
(d) g(x)=5x4+x2−3x−2