How Many Positive Real Solutions Does The Polynomial Equation $5x^3 + X^2 + 7x - 28 = 0$ Have?

The quest to understand polynomial equations is a cornerstone of algebra, and a key aspect of this understanding lies in determining the nature and number of their roots, also known as solutions or zeros. Among these roots, the positive real solutions hold particular significance in various applications across mathematics, science, and engineering. This article delves into the fascinating realm of polynomial equations, specifically focusing on how to determine the possible number of positive real solutions using Descartes' Rule of Signs. We will dissect a given polynomial equation, $5x^3 + x^2 + 7x - 28 = 0$, and employ this powerful rule to unravel the mystery of its positive real roots.

Descartes' Rule of Signs is a theorem that provides valuable information about the nature of a polynomial's roots, particularly the number of positive and negative real roots. This rule acts as a detective, offering clues about the possible solutions without explicitly solving the equation. It states two key principles:

1. The number of positive real roots of a polynomial equation $P(x) = 0$ is either equal to the number of sign changes in the coefficients of $P(x)$ or less than that by an even number.
2. The number of negative real roots is either equal to the number of sign changes in the coefficients of $P(-x)$ or less than that by an even number.

To effectively use Descartes' Rule of Signs, it is crucial to grasp the concept of "sign changes." A sign change occurs when we move from one term in the polynomial to the next and the sign of the coefficient changes (from positive to negative or vice versa). Constant terms with zero coefficients are ignored. It's also important to note the phrase "or less than that by an even number." This acknowledges the possibility of non-real (complex) roots, which always come in conjugate pairs, thereby reducing the number of real roots by even increments. Descartes' Rule of Signs helps to narrow down the possible number of real roots, making the process of finding the roots more efficient. Consider the polynomial $P(x) = x^5 - 3x^3 + 2x - 1$. To apply Descartes' Rule of Signs, we first count the sign changes in $P(x)$. There's a change from the first term ($x^5$) to the second term ($-3x^3$), another from the second term to the third term ($2x$), and a final change from the third term to the constant term ($-1$). That gives us a total of three sign changes. This implies that there could be three positive real roots or one positive real root (3 - 2 = 1). Next, we evaluate $P(-x) = (-x)^5 - 3(-x)^3 + 2(-x) - 1 = -x^5 + 3x^3 - 2x - 1$. There are two sign changes in $P(-x)$, indicating that there are either two negative real roots or no negative real roots (2 - 2 = 0). The rule doesn't give us the exact number, but it does provide a range of possibilities. This is especially useful because it helps to set expectations for the types of solutions to expect, informing the choice of methods for actually solving the equation. For example, if Descartes' Rule of Signs indicates the possibility of several real roots, numerical methods may be considered, or if the rule indicates fewer roots, it may be worthwhile to attempt factorization first. The power of Descartes' Rule of Signs is in its simplicity and the way it guides the solution process by providing initial insights into the nature of a polynomial's roots. Keep in mind, though, that the rule only gives potential numbers of roots; it doesn't locate the roots themselves. It is a tool that must be used in conjunction with other algebraic techniques to fully solve polynomial equations.

Let's apply Descartes' Rule of Signs to the given polynomial equation:

$5x^3 + x^2 + 7x - 28 = 0$

First, we count the sign changes in the coefficients of the polynomial as it is written. We start with the coefficient of the $x^3$ term, which is +5. The next term, $x^2$, has a coefficient of +1. There is no sign change here. The following term, $7x$, has a coefficient of +7. Again, there's no sign change. Finally, the constant term is -28. Here, we observe a sign change from +7 to -28. Therefore, there is only one sign change in the original polynomial. This tells us that there is exactly one positive real root, according to Descartes' Rule of Signs. To clarify this, let's illustrate the sequence of coefficients and the sign changes:

• +5 (coefficient of $x^3$)
• +1 (coefficient of $x^2$)
• +7 (coefficient of $x$)
• -28 (constant term)

As we move from +7 to -28, we encounter our single sign change. This definitive indication of one positive real root is a significant piece of information, especially when trying to solve the equation or understand the nature of its solutions. The beauty of Descartes' Rule of Signs is that it provides this insight without requiring us to actually solve the equation, which can be particularly beneficial for higher-degree polynomials where finding roots can be quite complex. By simply observing the pattern of signs in the coefficients, we gain a solid understanding of the number of potential positive real roots. However, it’s important to remember that Descartes' Rule of Signs only provides the possible number of positive real roots and doesn't specify their exact values. For that, one would need to apply other algebraic methods, numerical techniques, or graphing tools. The rule acts as a helpful guide, narrowing down the possibilities and shaping the approach to solving the equation more efficiently. The next step would be to consider the number of negative real roots and then assess any complex roots that might be present, which further enhances the overall understanding of the polynomial's characteristics.

To determine the possible number of negative real solutions, we need to analyze $P(-x)$. Substituting $-x$ for $x$ in the given polynomial equation, we get:

$P(-x) = 5(-x)^3 + (-x)^2 + 7(-x) - 28$

Simplifying this expression, we obtain:

$P(-x) = -5x^3 + x^2 - 7x - 28$

Now, let's count the sign changes in the coefficients of $P(-x)$. The coefficients are -5, +1, -7, and -28. Starting from -5, we observe a sign change to +1. From +1 to -7, there is another sign change. However, there is no sign change from -7 to -28. Thus, there are two sign changes in $P(-x)$. According to Descartes' Rule of Signs, this means that there could be two negative real roots or zero negative real roots (2 - 2 = 0). It's important to note that the number of negative real roots is either the number of sign changes or that number decreased by an even integer. This is because complex roots always come in conjugate pairs, thus reducing the number of real roots in pairs. Analyzing the negative solutions gives us additional insights into the behavior of the polynomial. Specifically, in the context of our original equation, the presence of either two or zero negative real roots, coupled with the single positive real root we found earlier, helps to shape a comprehensive picture of the polynomial's root structure. By knowing the possible range of negative real roots, we can better anticipate the number and nature of the solutions when attempting to solve the equation explicitly. This also guides the selection of appropriate methods for finding the roots, whether they are analytical techniques or numerical approximations. Moreover, analyzing P(-x) is essential because it completes the information Descartes' Rule of Signs provides, giving a more rounded view of all possible real roots, before considering the potential for complex roots. It's like examining different angles of a shape to fully understand its form, where the original polynomial gives the positive aspects, and $P(-x)$ sheds light on the negative ones. The entire analysis is critical for gaining a deep comprehension of the equation's solutions.

Based on our analysis using Descartes' Rule of Signs:

• There is exactly one positive real solution.
• There are either two or zero negative real solutions.

Combining these findings, we can deduce the possible scenarios for the types of roots of the polynomial equation $5x^3 + x^2 + 7x - 28 = 0$. The polynomial is of degree 3, which means it has a total of 3 roots (counting real and complex roots). We've determined that there is precisely one positive real root. If there are two negative real roots, then we have a total of three real roots (1 positive and 2 negative), meaning there are no complex roots in this scenario. On the other hand, if there are zero negative real roots, we still have the one positive real root. This leaves two roots unaccounted for. Since complex roots occur in conjugate pairs, the remaining two roots must be a pair of complex conjugate roots. Thus, the possible number of real solutions can be summarized as follows: either three real roots (one positive and two negative) or one real root (positive) and two complex roots. Understanding the nature and count of possible roots is critical in various applications, from engineering design to financial modeling. For instance, in a physical system modeled by a polynomial equation, the real roots might represent stable states, while complex roots might indicate oscillatory behavior. In economic models, real roots could represent equilibrium points, and complex roots might imply cyclical economic patterns. By delineating the possible number of real solutions, we not only solve a mathematical puzzle but also gain a tool for interpreting and predicting real-world phenomena. Furthermore, this analysis aids in the practical task of finding the solutions. Knowing that there's one positive real root, for example, directs the search toward positive values. Similarly, knowing the possibilities for negative and complex roots helps choose appropriate solution methods, such as numerical approximation techniques for real roots or algebraic methods for both real and complex roots.

In conclusion, by applying Descartes' Rule of Signs to the polynomial equation $5x^3 + x^2 + 7x - 28 = 0$, we have successfully determined that there is exactly one positive real solution. Furthermore, by analyzing $P(-x)$, we found that there are either two or zero negative real solutions. This comprehensive analysis provides valuable insights into the nature and number of real roots of the polynomial, guiding us toward a deeper understanding of its behavior. The power of Descartes' Rule of Signs lies in its ability to provide crucial information about the roots of a polynomial without requiring explicit solution. This makes it an indispensable tool in the arsenal of anyone studying algebra and related fields. Through this examination, we've not only answered the specific question about positive real solutions but also underscored the importance of understanding the underlying principles and theorems that govern polynomial equations. Looking back at the journey of applying Descartes' Rule, we recognize the simplicity and efficiency of the method. It offers a gateway to understanding the complex world of polynomial roots, where sign changes act as clues to unlock the mysteries hidden within the equations. This insight helps to make educated guesses about the nature and quantity of the roots, especially in preliminary analyses. The next step might involve using other methods such as the rational root theorem, graphing techniques, or numerical methods to find the roots explicitly. However, the groundwork provided by Descartes' Rule of Signs streamlines these subsequent steps. Ultimately, our exploration demonstrates how mastering such rules can empower us to tackle complex mathematical problems and appreciate the elegance and structure inherent in polynomial equations. The exploration deepens our understanding and adds a layer of predictability to what might otherwise seem like an abstract equation.

Therefore, the correct answer is A. One

Determine Positive Real Solutions for 5x^3 + x^2 + 7x - 28 = 0