Proving P Equals R When 2 And 1/2 Are Roots Of Px² + 5x + R

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Introduction

In this comprehensive article, we delve into the intricacies of quadratic equations and their roots. Specifically, we will explore the problem: Given that 2 and 1/2 are the zeros of the quadratic polynomial px² + 5x + r, prove that p = r. This problem elegantly combines the concepts of polynomial roots, the relationship between coefficients and roots, and algebraic manipulation. Understanding these concepts is crucial for anyone studying algebra and polynomial equations. Our discussion will cover the fundamental theorems involved, the step-by-step solution to the problem, and the underlying principles that make this proof work. This detailed exploration aims to provide a clear and thorough understanding of the topic, making it accessible to students and enthusiasts alike. To start with, let's define the basics and establish the framework for our proof.

The core idea revolves around the fact that if α and β are the zeros of a quadratic polynomial ax² + bx + c, then the sum of the zeros (α + β) is equal to -b/a, and the product of the zeros (αβ) is equal to c/a. These relationships are derived from Vieta's formulas, which are fundamental in polynomial theory. By applying these formulas to the given polynomial px² + 5x + r and the zeros 2 and 1/2, we can establish equations involving p and r. The subsequent algebraic manipulation of these equations will lead us to the desired conclusion, proving that p = r. This method not only solves the specific problem but also reinforces the understanding of how polynomial coefficients and roots are interconnected. Throughout this article, we will emphasize clarity and precision, ensuring that each step is well-explained and logically sound. By mastering this problem, you will gain a deeper appreciation for the elegance and power of algebraic reasoning.

This article also aims to provide a comprehensive understanding of polynomial roots and their relationship with coefficients. By dissecting the problem and explaining each step in detail, we aim to make the solution accessible to a wide audience, from students learning algebra to enthusiasts keen on deepening their mathematical knowledge. The methodical approach used here will serve as a valuable tool for tackling similar problems and enhancing problem-solving skills in mathematics. Moreover, we will discuss the broader implications of Vieta's formulas and their applications in various mathematical contexts. This will not only solidify the understanding of this particular problem but also provide a foundational knowledge base for more advanced topics in algebra and polynomial theory. Let's embark on this mathematical journey to unravel the proof and understand the beauty of algebraic relationships.

Understanding the Problem

The problem at hand is a classic example of how the roots of a polynomial are related to its coefficients. The given quadratic polynomial is px² + 5x + r, and we know that 2 and 1/2 are its zeros. A zero of a polynomial is a value that, when substituted for the variable x, makes the polynomial equal to zero. In other words, if α is a zero of a polynomial f(x), then f(α) = 0. This fundamental concept is the cornerstone of our approach.

Since 2 and 1/2 are zeros of px² + 5x + r, substituting these values into the polynomial should yield zero. This gives us two equations: p(2)² + 5(2) + r = 0 and p(1/2)² + 5(1/2) + r = 0. These equations form the basis for our proof. By manipulating these equations, we can establish a relationship between p and r. This is where the power of algebraic manipulation comes into play. We will simplify these equations, isolate variables, and look for common terms that can help us equate p and r. The initial substitution is a crucial step, transforming the problem from a theoretical statement into concrete algebraic equations that we can work with.

Moreover, the structure of the quadratic polynomial itself provides additional insights. The coefficients p and r, along with the constant term 5, play a vital role in determining the behavior and properties of the polynomial. The relationship between these coefficients and the zeros is governed by Vieta's formulas, which we will utilize later in the proof. Understanding this connection is key to solving the problem efficiently. For example, the coefficient of the x term (5 in this case) is related to the sum of the roots, while the constant term (r) is related to the product of the roots. By leveraging these relationships, we can bypass direct substitution and arrive at the solution more elegantly. However, for the sake of clarity and thoroughness, we will initially focus on the substitution method to build a strong foundation. The combination of direct substitution and Vieta's formulas will provide a comprehensive understanding of the problem and its solution. Let's now proceed with the substitution and simplification process to uncover the relationship between p and r.

Applying the Zero Property

As mentioned earlier, the cornerstone of our solution lies in the fact that if 2 and 1/2 are zeros of the polynomial px² + 5x + r, then substituting these values for x will result in the polynomial evaluating to zero. This is a direct application of the definition of a polynomial zero. Let's proceed with this substitution step by step.

First, we substitute x = 2 into the polynomial px² + 5x + r: p(2)² + 5(2) + r = 0 Simplifying this equation, we get: 4p + 10 + r = 0. This is our first equation, which establishes a linear relationship between p and r. It's a crucial step because it transforms the polynomial condition into a manageable algebraic form. The equation 4p + 10 + r = 0 tells us that there's a specific balance between p and r that satisfies the polynomial's zero property when x = 2. This balance is what we aim to uncover and relate to the condition when x = 1/2.

Next, we substitute x = 1/2 into the polynomial px² + 5x + r: p(1/2)² + 5(1/2) + r = 0 Simplifying this equation, we get: p(1/4) + 5/2 + r = 0. To eliminate the fraction, we multiply the entire equation by 4, resulting in: p + 10 + 4r = 0. This is our second equation, another linear relationship between p and r. Just like the first equation, it represents a condition that must be satisfied for 1/2 to be a zero of the polynomial. Comparing this equation with the first one, we see that the coefficients of p and r are different, but the constant term remains the same. This observation is a key clue that will help us in the next step, where we will manipulate these equations to isolate p and r and ultimately prove that they are equal.

These two equations, 4p + 10 + r = 0 and p + 10 + 4r = 0, are the fruits of our initial substitutions. They encapsulate the information that 2 and 1/2 are zeros of the polynomial. Now, the challenge is to extract the desired result (p = r) from these equations. We will employ algebraic techniques such as subtraction or substitution to eliminate one variable and solve for the other. This methodical approach, starting from the basic definition of polynomial zeros and progressing through algebraic manipulation, demonstrates the power of mathematical reasoning. Let's move on to the next section where we will solve these equations and complete the proof.

Solving the Equations

Now that we have established two equations, 4p + 10 + r = 0 and p + 10 + 4r = 0, the next step is to solve them simultaneously to find the relationship between p and r. There are several methods to solve a system of linear equations, such as substitution, elimination, and matrix methods. In this case, the elimination method is particularly effective due to the presence of a common constant term (+10) in both equations.

To use the elimination method, we aim to subtract one equation from the other in such a way that one of the variables is eliminated. Looking at our equations, we can see that subtracting the second equation from the first will eliminate the constant term (+10). This simplification is crucial because it reduces the complexity of the equations and brings us closer to isolating p and r. The process of elimination is a fundamental technique in algebra, allowing us to systematically solve for unknowns by strategically manipulating equations.

Subtracting the second equation (p + 10 + 4r = 0) from the first equation (4p + 10 + r = 0), we get: (4p + 10 + r) - (p + 10 + 4r) = 0 - 0 Simplifying this, we have: 4p + 10 + r - p - 10 - 4r = 0 Combining like terms, we get: 3p - 3r = 0. This equation is significantly simpler than our initial equations. The elimination of the constant term has resulted in a direct relationship between p and r. The equation 3p - 3r = 0 is a linear equation in two variables, but it's special because it immediately reveals the connection we are looking for. Dividing both sides of the equation by 3, we further simplify it to: p - r = 0. This equation is the key to our proof. It states that the difference between p and r is zero, which implies that p and r are equal. Adding r to both sides of the equation, we finally arrive at: p = r. This completes our proof. By systematically applying algebraic techniques, we have shown that if 2 and 1/2 are the zeros of the polynomial px² + 5x + r, then p must be equal to r. The elegance of this proof lies in its simplicity and the clear, logical steps that lead to the conclusion. From the initial substitution to the final equation, each step builds upon the previous one, demonstrating the power of algebraic reasoning. Let's now reflect on the implications of this result and the broader context of polynomial theory.

Conclusion: Proving p = r

In conclusion, we have successfully demonstrated that if 2 and 1/2 are the zeros of the quadratic polynomial px² + 5x + r, then p = r. This proof elegantly combines the definition of polynomial zeros with algebraic manipulation techniques to arrive at a concise and compelling result. The journey began with understanding the problem, followed by substituting the zeros into the polynomial, and culminated in solving the resulting equations to establish the equality of p and r.

Our method involved substituting x = 2 and x = 1/2 into the given polynomial, which yielded two equations: 4p + 10 + r = 0 and p + 10 + 4r = 0. We then employed the elimination method to solve these equations simultaneously. By subtracting the second equation from the first, we eliminated the constant term and simplified the equation to 3p - 3r = 0. This pivotal step directly revealed the relationship between p and r. Dividing both sides by 3, we obtained p - r = 0, which immediately implies that p = r. This final equation is the crux of our proof, showcasing the direct and logical connection between the zeros of the polynomial and its coefficients.

The significance of this proof extends beyond the specific problem. It highlights the fundamental relationship between the roots and coefficients of a polynomial, a cornerstone concept in algebra. Understanding this relationship is crucial for solving a wide range of problems involving polynomial equations. Moreover, the techniques used in this proof, such as substitution and elimination, are widely applicable in various mathematical contexts. The methodical approach we followed, starting from basic definitions and progressing through logical steps, is a valuable problem-solving strategy that can be applied to other mathematical challenges.

This exploration also underscores the power of algebraic manipulation in unraveling mathematical truths. The seemingly complex problem was reduced to a simple equation through strategic algebraic steps. This demonstrates that with a clear understanding of underlying principles and the application of appropriate techniques, complex problems can be simplified and solved. Furthermore, this exercise reinforces the importance of precision and attention to detail in mathematics. Each step in the proof was carefully executed to ensure the validity of the final result.

In summary, the proof that p = r when 2 and 1/2 are the zeros of px² + 5x + r is a testament to the elegance and power of mathematical reasoning. It not only solves a specific problem but also provides valuable insights into the broader concepts of polynomial theory and algebraic techniques. This understanding will undoubtedly benefit students and enthusiasts in their further exploration of mathematics.

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