Questions: Which of the following could be the complete chart of possible types and numbers of the roots of the function F(x)=-2x^5+3x^3+2x^2-x-3 ? Positive Negative Imaginary 2 3 0 2 1 2 Positive Negative Imaginary 2 3 0 2 1 2 0 3 2 Positive Negative Imaginary 2 2 1 2 0 3 0 2 3 0 0 5 Positive Negative Imaginary 2 3 0 2 1 2 0 3 2 0 1 4

Which of the following could be the complete chart of possible types and numbers of the roots of the function F(x)=-2x^5+3x^3+2x^2-x-3 ?

Positive Negative Imaginary
2 3 0
2 1 2

Positive Negative Imaginary
2 3 0
2 1 2
0 3 2

Positive Negative Imaginary
2 2 1
2 0 3
0 2 3
0 0 5

Positive Negative Imaginary
2 3 0
2 1 2
0 3 2
0 1 4

Transcript text: Which of the following could be the complete chart of possible types and numbers of the roots of the function $F(x)=-2 x^{5}+3 x^{3}+2 x^{2}-x-3$ ? \begin{tabular}{|c|c|c|} \hline Positive & Negative & Imaginary \\ \hline 2 & 3 & 0 \\ \hline 2 & 1 & 2 \\ \hline \end{tabular} \begin{tabular}{|c|c|c|} \hline Positive & Negative & Imaginary \\ \hline 2 & 3 & 0 \\ \hline 2 & 1 & 2 \\ \hline 0 & 3 & 2 \\ \hline \end{tabular} \begin{tabular}{|c|c|c|} \hline Positive & Negative & Imaginary \\ \hline 2 & 2 & 1 \\ \hline 2 & 0 & 3 \\ \hline 0 & 2 & 3 \\ \hline 0 & 0 & 5 \\ \hline \end{tabular} \begin{tabular}{|c|c|c|} \hline Positive & Negative & Imaginary \\ \hline 2 & 3 & 0 \\ \hline 2 & 1 & 2 \\ \hline 0 & 3 & 2 \\ \hline 0 & 1 & 4 \\ \hline \end{tabular}

Solution

Solution Steps

To determine the possible types and numbers of roots for the polynomial $ F(x) = -2x^5 + 3x^3 + 2x^2 - x - 3 $, we can use Descartes' Rule of Signs. This rule helps us find the number of positive and negative real roots by analyzing the sign changes in the polynomial and its transformation. Additionally, the Fundamental Theorem of Algebra tells us that the total number of roots (including complex roots) is equal to the degree of the polynomial, which is 5 in this case.

Positive Roots: Count the sign changes in $ F(x) $.
Negative Roots: Count the sign changes in $ F(-x) $.
Imaginary Roots: Use the total degree of the polynomial to determine the remaining roots after accounting for real roots.

Step 1: Determine Positive Roots

Using Descartes' Rule of Signs, we analyze the polynomial $ F(x) = -2x^5 + 3x^3 + 2x^2 - x - 3 $. The number of sign changes in $ F(x) $ is 2, indicating that there are either 2 or 0 positive real roots.

Step 2: Determine Negative Roots

Next, we evaluate $ F(-x) = 2x^5 - 3x^3 + 2x^2 + x - 3 $. The number of sign changes in $ F(-x) $ is 3, which suggests there are either 3, 1, or no negative real roots.

Step 3: Determine Imaginary Roots

The degree of the polynomial $ F(x) $ is 5. The total number of roots (including complex roots) must equal the degree. Given the findings from the previous steps, we have:

Positive roots: 2
Negative roots: 3

Thus, the number of imaginary roots can be calculated as: \[ \text{Imaginary Roots} = \text{Degree} - (\text{Positive Roots} + \text{Negative Roots}) = 5 - (2 + 3) = 0 \]