- Identify the coefficients of \(Q(x) = 5x^4 + 0x^3 + 6x^2 - 3x + 3\) as \(5, 6, -3, 3\).
- Observe 2 sign changes in the sequence \(+, +, -, +\), implying 2 or 0 positive real zeros.
- Find \(Q(-x) = 5x^4 + 6x^2 + 3x + 3\) and observe 0 sign changes in the sequence \(+, +, +, +\), implying 0 negative real zeros.
- Final result: Positive roots \(\boxed{2, 0}\), negative roots \(\boxed{0}\).
Explanation
- Analysis of the polynomial coefficients for positive roots
We are given the polynomial \(Q(x) = 5x^4 + 6x^2 - 3x + 3\). Our objective is to determine the possible number of positive and negative real zeros using Descartes' Rule of Signs. To analyze the positive real zeros, we examine the sign changes in the sequence of coefficients of \(Q(x)\). Writing the polynomial in standard form to include all powers of \(x\), we have \(Q(x) = 5x^4 + 0x^3 + 6x^2 - 3x + 3\). The coefficients are \(5, 0, 6, -3, 3\). Ignoring the zero coefficient, the sequence of signs is \(+, +, -, +\).
- Determining positive real zeros
We observe the sign changes in the sequence \(+, +, -, +\):
- From the first \(+\) to the second \(+\), there is no change.
- From the second \(+\) to the \(-\), there is one change.
- From the \(-\) to the final \(+\), there is one change.
This gives us a total of two sign changes. According to Descartes' Rule of Signs, the number of positive real zeros is equal to the number of sign changes (2) or less than it by an even number (2 - 2 = 0). Thus, the possible number of positive real zeros is 2 or 0.
- Analysis of the polynomial coefficients for negative roots
To analyze the negative real zeros, we examine the sign changes in the sequence of coefficients of \(Q(-x)\). We substitute \(-x\) for \(x\) in the polynomial:
\[Q(-x) = 5(-x)^4 + 0(-x)^3 + 6(-x)^2 - 3(-x) + 3\]
\[Q(-x) = 5x^4 + 0x^3 + 6x^2 + 3x + 3\]
The coefficients of \(Q(-x)\) are \(5, 0, 6, 3, 3\). Ignoring the zero coefficient, the sequence of signs is \(+, +, +, +\).
- Determining negative real zeros
In the sequence \(+, +, +, +\), there are zero sign changes. According to Descartes' Rule of Signs, the number of negative real zeros is equal to the number of sign changes (0), or less than it by an even number. Since 0 is already the smallest non-negative integer possible, the only possibility is that there are 0 negative real zeros.
- Conclusion
Following our analysis, the possible numbers of positive real zeros are 2 or 0, and the only possible number of negative real zeros is 0.
APossible number(s) of positive real zeros: \(\boxed{2, 0}\)
BPossible number(s) of negative real zeros: \(\boxed{0}\)
Examples
Descartes' Rule of Signs is a powerful tool in engineering and computer science, especially when analyzing the stability of systems. In control theory, for instance, a polynomial's coefficients describe the characteristics of a system; determining the number of roots (poles) in different parts of the complex plane (or specifically, identifying positive or negative real roots) helps determine if a physical system will remain stable or diverge. By simply looking at the sign pattern of the polynomial coefficients, an engineer can quickly narrow down the number of positive or negative roots without needing to compute the exact roots numerically.