Example 8 from Algebra and Trigonometry, 5.5 Zeros of Polynomial Functions

Using Descartes’ Rule of Signs

Use Descartes’ Rule of Signs to determine the possible numbers of positive and negative real zeros for $f (x) = - x^{4} - 3 x^{3} + 6 x^{2} - 4 x - 12.$

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
Begin by determining the number of sign changes.

Figure 3

There are two sign changes, so there are either 2 or 0 positive real roots. Next, we examine $f (- x)$ to determine the number of negative real roots.

$\begin{matrix} f (- x) & = & - {(- x)}^{4} - 3 {(- x)}^{3} + 6 {(- x)}^{2} - 4 (- x) - 12 \\ f (- x) & = & - x^{4} + 3 x^{3} + 6 x^{2} + 4 x - 12 \end{matrix}$

Figure 4

Again, there are two sign changes, so there are either 2 or 0 negative real roots.

There are four possibilities, as we can see in Table 1.

Positive Real Zeros Negative Real Zeros Complex Zeros Total Zeros

2 2 0 4

2 0 2 4

0 2 2 4

0 0 4 4

Table 1

Positive Real Zeros	Negative Real Zeros	Complex Zeros	Total Zeros
2	2	0	4
2	0	2	4
0	2	2	4
0	0	4	4

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Use Descartes’ Rule of Signs to determine the maximum possible numbers of positive and negative real zeros for $f (x) = 2 x^{4} - 10 x^{3} + 11 x^{2} - 15 x + 12.$ Use a graph to verify the numbers of positive and negative real zeros for the function.

How did it go?

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Solution

Your turn

Solution

Your turn