Exercise 59 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)= -1x^4- 3x^3+8x^2+2x- 14 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)=-(- x)^4-3(- x)^3+8(- x)^2+2(- x)-14 ⇕ P(- x)= - 1x^4+3x^3+8x^2- 2x- 14 We can see above that there are two sign changes, ( -) to (+) and (+) to ( -). Therefore, there are either 2 or 0 negative real zeros.