Exercise 34 Page 203

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

• The number of positive real zeros of $f$ is either equal to the number of sign changes between consecutive coefficients of $f(x)$ or is less than that by an even number.
• The number of negative real zeros of $f$ is either equal to the number of sign changes between consecutive coefficients of $f(\text{-}x)$ or is less than that by an even number.

Positive Real Zeros

Consider the given polynomial function $g(x).$ We can see above that there is one sign change, $(-)$ to $(+).$ Therefore, there is $1$ positive real zero.

Negative Real Zeros

Now consider $g(\text{-}x).$ We can see that in this case there is no sign change. Thus, there is zero negative real zeros.

Imaginary Zeros

To calculate the possible number of imaginary zeros, we first need to find the total number of complex zeros (both real and imaginary). According to the Fundamental Theorem of Algebra, the number of complex zeros is given by the degree of the polynomial. In this case, there are $3$ complex zeros. The difference between this number and the number of real roots — positive and negative — is the number of imaginary zeros. We already found these numbers above, so let's calculate the possible number of imaginary zeros.

Total Number of Zeros	Number of $\pm$ Real Zeros	Number of Imaginary Zeros
$3$	$1+0=1$	$3-1=2$

The given polynomial has $2$ imaginary zeros.