Determining Positive Real Solutions For Polynomial Equations 5x^3 + X^2 + 7x - 28 = 0

In the realm of algebra, polynomial equations hold a significant position, and unraveling their solutions is a fundamental task. Among the various types of solutions, real solutions are of particular interest, especially the positive real solutions. These solutions represent points where the polynomial function intersects the positive x-axis, offering valuable insights into the function's behavior and characteristics. In this article, we embark on a journey to explore the methods and techniques used to determine the possible number of positive real solutions for a given polynomial equation. Let's delve into the intricacies of Descartes' Rule of Signs and its application in solving the equation:

5x³ + x² + 7x - 28 = 0

Descartes' Rule of Signs: A Guiding Light

At the heart of our exploration lies Descartes' Rule of Signs, a powerful theorem that provides valuable information about the nature of polynomial equations' roots. This rule establishes a connection between the number of sign changes in the polynomial's coefficients and the possible number of positive and negative real roots. Specifically, it states:

1. The number of positive real roots of a polynomial equation is either equal to the number of sign changes in the coefficients 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 utilize Descartes' Rule of Signs, we must first understand the concept of sign changes in the coefficients. A sign change occurs whenever two consecutive coefficients have opposite signs. For instance, in the polynomial equation:

5x³ + x² + 7x - 28 = 0

We observe one sign change between the positive coefficient of 7x and the negative constant term -28. This indicates that the equation has either one positive real root or no positive real roots (1 - 2 = -1, which is not a valid number of roots, so we consider 1 - 0 = 1).

Applying Descartes' Rule to the Equation

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

5x³ + x² + 7x - 28 = 0

1. Positive Real Roots:

   As we previously identified, there is one sign change in the coefficients (from +7 to -28). Therefore, according to Descartes' Rule of Signs, there is exactly one positive real root.

2. Negative Real Roots:

   To determine the possible number of negative real roots, we need to consider P(-x). Substituting -x for x in the equation, we get:

   5(-x)³ + (-x)² + 7(-x) - 28 = 0

   Simplifying, we have:

   -5x³ + x² - 7x - 28 = 0

   In this transformed equation, we observe two sign changes: from -5 to +1 and from +1 to -7. Therefore, there are either two or zero negative real roots.

Delving Deeper: The Nature of Roots

Descartes' Rule of Signs provides a powerful tool for narrowing down the possibilities for the number of positive and negative real roots. However, it's crucial to remember that the rule only gives us the possible number of roots. To gain a more complete understanding of the nature of the roots, we can employ additional techniques, such as the Rational Root Theorem and graphical analysis.

The Rational Root Theorem

The Rational Root Theorem helps us identify potential rational roots of a polynomial equation. It states that if a polynomial equation has integer coefficients, then any rational root must be of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. For our equation, the constant term is -28, and the leading coefficient is 5. Therefore, the possible rational roots are ±1, ±2, ±4, ±7, ±14, ±28, ±1/5, ±2/5, ±4/5, ±7/5, ±14/5, and ±28/5.

Graphical Analysis

Graphing the polynomial function can provide valuable insights into the nature of its roots. The points where the graph intersects the x-axis represent the real roots of the equation. By analyzing the graph, we can visually confirm the number of positive and negative real roots and their approximate values. For the equation 5x³ + x² + 7x - 28 = 0, the graph intersects the x-axis at one positive point, confirming our earlier finding that there is one positive real root.

The Significance of Real Solutions

Real solutions, particularly positive real solutions, hold significant importance in various fields of mathematics, science, and engineering. They represent tangible and physically meaningful values in real-world applications. For instance, in physics, positive real solutions might represent the distance, time, or velocity of an object. In economics, they could represent the price, quantity, or profit of a commodity. Understanding the nature and number of positive real solutions allows us to make informed decisions and predictions in these diverse domains.

Conclusion: One Positive Real Solution Unveiled

By employing Descartes' Rule of Signs, we have successfully determined that the polynomial equation 5x³ + x² + 7x - 28 = 0 has exactly one positive real solution. This solution represents a point where the polynomial function intersects the positive x-axis, providing valuable information about the function's behavior. While Descartes' Rule of Signs serves as a powerful tool for narrowing down the possibilities, additional techniques like the Rational Root Theorem and graphical analysis can further enhance our understanding of the nature of polynomial equation roots. The quest for understanding polynomial solutions is an ongoing journey, with each technique and theorem contributing to a more comprehensive and insightful understanding.

Therefore, the correct answer is C. One.