Exercise 57 Page 350

We can use Descartes' Rule of Signs to learn about the number of positive and negative real zeros for the given polynomial function. Let P(x) be a polynomial with real coefficients written in standard form.

• The number of positive real roots of P(x)=0 is either equal to the number of sign changes between consecutive coefficients of P(x) or is less than that by an even number.
• The number of negative real roots of P(x)=0 is either equal to the number of sign changes between consecutive coefficients of P(- x) or is less than that by an even number.

Positive Real Zeros

Consider the given polynomial P(x). P(x)= - 3x^3+11x^2+12x- 8 We can see above that there are two sign changes, ( -) to (+) and (+) to ( -). Therefore, there are either 2 or 0 positive real zeros.

Negative Real Zeros

Now consider P(- x). P(- x)=- 3(- x)^3+11(- x)^2+12(- x)-8 ⇕ P(- x)=3x^3+11x^2- 12x- 8 We can see that there is only one sign change, (+) to ( -). Thus, there is only one negative real zeros.