MAT1511/101/0/2025QUESTION 22.1 Use Descartes' Rule Of Signs To Describe All Possibilities For The Number Of Positive, Negative, And Imaginary Zeros Of $P(x$\], Where $P(x) = 4x^3 - 3x^2 - 7x + 9$. (You May Summarize Your Answer In

Introduction

In mathematics, Descartes' Rule of Signs is a powerful tool used to determine the number of positive, negative, and imaginary zeros of a polynomial function. This rule is based on the observation that the number of sign changes in the coefficients of a polynomial is equal to the number of positive real zeros, or is less than that number by a positive even integer. In this article, we will explore the application of Descartes' Rule of Signs to the polynomial function $P(x) = 4x^3 - 3x^2 - 7x + 9$.

Descartes' Rule of Signs

Descartes' Rule of Signs is a simple yet effective method for determining the number of zeros of a polynomial function. The rule states that the number of sign changes in the coefficients of a polynomial is equal to the number of positive real zeros, or is less than that number by a positive even integer. To apply this rule, we need to examine the coefficients of the polynomial and count the number of sign changes.

Applying Descartes' Rule of Signs to $P(x)$

Let's apply Descartes' Rule of Signs to the polynomial function $P(x) = 4x^3 - 3x^2 - 7x + 9$. We can see that there are three sign changes in the coefficients of this polynomial: from $4$ to $-3$, from $-3$ to $-7$, and from $-7$ to $9$. Therefore, according to Descartes' Rule of Signs, the number of positive real zeros of $P(x)$ is either $3$ or $1$.

Determining the Number of Negative Zeros

To determine the number of negative zeros of $P(x)$, we need to apply Descartes' Rule of Signs to the polynomial function $P(-x)$. We can obtain $P(-x)$ by substituting $-x$ for $x$ in the original polynomial function. This gives us $P(-x) = -4x^3 - 3x^2 + 7x + 9$. We can see that there is only one sign change in the coefficients of this polynomial: from $-4$ to $-3$. Therefore, according to Descartes' Rule of Signs, the number of negative real zeros of $P(x)$ is either $1$ or $3$.

Determining the Number of Imaginary Zeros

To determine the number of imaginary zeros of $P(x)$, we need to consider the degree of the polynomial. Since the degree of $P(x)$ is $3$, which is an odd number, we know that there must be at least one imaginary zero. In fact, the number of imaginary zeros is equal to the difference between the degree of the polynomial and the number of real zeros. Therefore, if we assume that there are $3$ positive real zeros and $1$ negative real zero, we can conclude that there is $1$ imaginary zero.

Conclusion

In conclusion, we have applied Descartes' Rule of Signs to the polynomial function $P(x) = 4x^3 - 3x^2 - 7x + 9$ and determined the possibilities for the number of positive, negative, and imaginary zeros. We have found that the number of positive real zeros is either $3$ or $1$, the number of negative real zeros is either $1$ or $3$, and the number of imaginary zeros is $1$. This comprehensive analysis provides valuable insights into the behavior of the polynomial function and its zeros.

Possible Scenarios

Based on our analysis, we can summarize the possible scenarios for the number of positive, negative, and imaginary zeros of $P(x)$ as follows:

• Scenario 1: $3$ positive real zeros, $1$ negative real zero, and $1$ imaginary zero.
• Scenario 2: $1$ positive real zero, $3$ negative real zeros, and $1$ imaginary zero.

These scenarios provide a comprehensive understanding of the behavior of the polynomial function and its zeros.

Implications

The implications of our analysis are significant, as they provide valuable insights into the behavior of the polynomial function and its zeros. By understanding the number of positive, negative, and imaginary zeros, we can gain a deeper appreciation for the properties of the polynomial function and its applications in various fields.

Future Directions

Q&A: Frequently Asked Questions about Descartes' Rule of Signs

Q: What is Descartes' Rule of Signs?

A: Descartes' Rule of Signs is a mathematical rule used to determine the number of positive, negative, and imaginary zeros of a polynomial function. It is based on the observation that the number of sign changes in the coefficients of a polynomial is equal to the number of positive real zeros, or is less than that number by a positive even integer.

Q: How do I apply Descartes' Rule of Signs to a polynomial function?

A: To apply Descartes' Rule of Signs, you need to examine the coefficients of the polynomial and count the number of sign changes. If the polynomial has a degree of $n$, then the number of sign changes is equal to the number of positive real zeros, or is less than that number by a positive even integer.

Q: What is the significance of the number of sign changes in a polynomial?

A: The number of sign changes in a polynomial is significant because it determines the number of positive real zeros of the polynomial. If the number of sign changes is equal to the degree of the polynomial, then the polynomial has the same number of positive real zeros as the degree of the polynomial. If the number of sign changes is less than the degree of the polynomial by a positive even integer, then the polynomial has fewer positive real zeros than the degree of the polynomial.

Q: How do I determine the number of negative zeros of a polynomial function?

A: To determine the number of negative zeros of a polynomial function, you need to apply Descartes' Rule of Signs to the polynomial function $P(-x)$. This involves substituting $-x$ for $x$ in the original polynomial function and counting the number of sign changes in the coefficients of the resulting polynomial.

Q: What is the relationship between the number of real zeros and the number of imaginary zeros of a polynomial function?

A: The number of imaginary zeros of a polynomial function is equal to the difference between the degree of the polynomial and the number of real zeros. If the polynomial has a degree of $n$ and $r$ real zeros, then the number of imaginary zeros is $n-r$.

Q: Can Descartes' Rule of Signs be used to determine the number of complex zeros of a polynomial function?

A: Yes, Descartes' Rule of Signs can be used to determine the number of complex zeros of a polynomial function. If the polynomial has a degree of $n$, then the number of complex zeros is equal to the number of sign changes in the coefficients of the polynomial, or is less than that number by a positive even integer.

Q: What are the limitations of Descartes' Rule of Signs?

A: The limitations of Descartes' Rule of Signs are that it only provides information about the number of zeros of a polynomial function, and does not provide information about the location or nature of the zeros. Additionally, the rule only applies to polynomials with real coefficients, and does not apply to polynomials with complex coefficients.

Q: How can Descartes' Rule of Signs be used in real-world applications?

A: Descartes' Rule of Signs can be used in a variety of real-world applications, including engineering, physics, and computer science. For example, the rule can be used to determine the stability of a system, or to analyze the behavior of a complex system.

Q: What are some common mistakes to avoid when applying Descartes' Rule of Signs?

A: Some common mistakes to avoid when applying Descartes' Rule of Signs include:

• Failing to count the number of sign changes in the coefficients of the polynomial.
• Failing to consider the degree of the polynomial when applying the rule.
• Failing to account for the number of real zeros when determining the number of imaginary zeros.
• Failing to consider the limitations of the rule, such as its applicability only to polynomials with real coefficients.