Finding Quadratic Polynomial From Zeroes A Step-by-Step Guide

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Introduction

In mathematics, specifically in algebra, a quadratic polynomial is a polynomial of degree two. It has the general form ax² + bx + c, where a, b, and c are constants and a ≠ 0. The zeroes of a quadratic polynomial are the values of x for which the polynomial equals zero. These zeroes are also known as the roots of the quadratic equation formed by setting the polynomial equal to zero. Given the zeroes of a quadratic polynomial, we can construct the polynomial itself. This article will delve into the process of finding a quadratic polynomial when its zeroes are known, providing a detailed explanation and a step-by-step approach to solve the problem at hand. We will explore the relationship between the zeroes and the coefficients of the quadratic polynomial and illustrate the method with a specific example. Understanding this concept is crucial for various applications in mathematics, physics, and engineering, where quadratic equations and polynomials frequently appear. By mastering this technique, you will be able to tackle similar problems with confidence and gain a deeper understanding of the fundamental properties of quadratic functions. The ability to construct a quadratic polynomial from its zeroes is a fundamental skill in algebra, with wide-ranging applications in various fields. This skill allows us to model real-world situations, solve optimization problems, and analyze the behavior of systems that can be described by quadratic equations. In this article, we will not only focus on the mechanics of constructing the polynomial but also emphasize the underlying concepts and their significance in a broader mathematical context. We will explore the connection between the zeroes of a polynomial and its factors, the role of the leading coefficient, and the uniqueness of the quadratic polynomial under certain conditions.

Understanding the Relationship Between Zeroes and Polynomials

The zeroes of a polynomial are the values of the variable (usually x) that make the polynomial equal to zero. For a quadratic polynomial ax² + bx + c, if α and β are the zeroes, then the polynomial can be expressed in the factored form as a(x - α)(x - β), where a is a non-zero constant. This fundamental relationship between zeroes and polynomials is the cornerstone of our approach. It allows us to reconstruct the polynomial if we know its zeroes and a scaling factor. The constant a plays a crucial role in determining the specific quadratic polynomial, as multiplying the entire polynomial by a constant does not change its zeroes but affects the coefficients. This understanding is essential for solving problems where we need to find a quadratic polynomial with specific zeroes and potentially additional constraints, such as a particular value at a given point. The factored form of the quadratic polynomial provides valuable insights into its behavior and properties. For instance, it directly reveals the x-intercepts of the graph of the polynomial, which are the points where the graph crosses the x-axis. It also allows us to easily determine the sign of the polynomial in different intervals, which is useful for solving inequalities and analyzing the function's behavior. Furthermore, the factored form can be used to derive the standard form ax² + bx + c by expanding the product, revealing the relationship between the zeroes and the coefficients b and c. This connection is captured by Vieta's formulas, which state that the sum of the zeroes is -b/a and the product of the zeroes is c/a. These formulas provide a powerful tool for verifying the correctness of our constructed polynomial and for solving problems where we need to find the zeroes given the coefficients.

Constructing a Quadratic Polynomial from its Zeroes

Given the zeroes α and β, the quadratic polynomial can be written in the form k(x - α)(x - β), where k is any non-zero constant. Expanding this expression, we get k(x² - (α + β)x + αβ). This form highlights the relationship between the zeroes and the coefficients of the polynomial. The coefficient of the x term is the negative of the sum of the zeroes, and the constant term is the product of the zeroes. The constant k acts as a scaling factor, allowing us to generate a family of quadratic polynomials with the same zeroes. This is because multiplying a polynomial by a constant does not change its roots. However, the specific value of k can be determined if we have additional information, such as a point that the polynomial passes through. This information provides an additional equation that allows us to solve for k. In many cases, we are interested in finding the simplest quadratic polynomial with the given zeroes, which corresponds to setting k = 1. However, in certain applications, a different value of k may be required to satisfy specific conditions. For example, we may need to find a quadratic polynomial with integer coefficients, in which case we would choose k to eliminate any fractions in the expression. The process of constructing a quadratic polynomial from its zeroes involves several key steps. First, we identify the given zeroes, α and β. Second, we form the factors (x - α) and (x - β). Third, we multiply these factors together to obtain the quadratic expression (x - α)(x - β). Finally, we multiply the expression by a constant k to account for the scaling factor. This constant can be chosen based on additional information or constraints, or it can be set to 1 for the simplest polynomial.

Solving the Specific Problem

In this case, the zeroes are given as 2/5 and -1/5. Let α = 2/5 and β = -1/5. We can construct the quadratic polynomial as follows:

  1. Start with the factored form: k(x - α)(x - β)
  2. Substitute the values of α and β: k(x - 2/5)(x + 1/5)
  3. Expand the expression inside the parentheses: k(x² + (1/5)x - (2/5)x - (2/25)) which simplifies to k(x² - (1/5)x - (2/25)).
  4. To eliminate fractions, we can choose k = 25, which gives us: 25*(x² - (1/5)x - (2/25)) = 25x² - 5x - 2.

Therefore, the quadratic polynomial is 25x² - 5x - 2. This solution demonstrates the practical application of the method described earlier. By substituting the given zeroes into the factored form and expanding the expression, we obtained a quadratic polynomial with the desired roots. The choice of k = 25 was crucial for eliminating fractions and obtaining a polynomial with integer coefficients. This step highlights the importance of considering the desired form of the polynomial when selecting the scaling factor. In this specific problem, the given answer choices are all quadratic polynomials with integer coefficients, which suggests that we need to choose a value of k that eliminates fractions. However, in other problems, we may be interested in finding a quadratic polynomial with specific coefficients or satisfying other constraints. In such cases, the choice of k may be different. The process of solving this problem also reinforces the connection between the zeroes of a polynomial and its coefficients. The sum of the zeroes, 2/5 + (-1/5) = 1/5, is related to the coefficient of the x term, which is -5/25 = -1/5. The product of the zeroes, (2/5) * (-1/5) = -2/25, is related to the constant term, which is -2/25. These relationships are consistent with Vieta's formulas, which provide a powerful tool for verifying the correctness of our solution.

Conclusion

We have successfully constructed a quadratic polynomial given its zeroes by utilizing the relationship between the zeroes and the factored form of the polynomial. The key steps involved substituting the zeroes into the factored form, expanding the expression, and choosing an appropriate constant to eliminate fractions or satisfy other constraints. This method provides a systematic approach for solving problems where we need to find a quadratic polynomial with specific zeroes. Understanding this process is fundamental for various applications in mathematics and other fields. The ability to construct a polynomial from its zeroes is a valuable skill that allows us to model real-world situations, solve equations, and analyze the behavior of functions. By mastering this technique, you will be well-equipped to tackle a wide range of problems involving quadratic polynomials and their properties. Furthermore, this process reinforces the connection between the zeroes and the coefficients of a polynomial, as captured by Vieta's formulas. These formulas provide a powerful tool for verifying the correctness of our solutions and for solving problems where we need to find the zeroes given the coefficients. In summary, the construction of a quadratic polynomial from its zeroes involves a combination of algebraic manipulation, conceptual understanding, and problem-solving skills. By practicing this technique and exploring its applications, you will gain a deeper appreciation for the beauty and power of mathematics.

Therefore, the correct answer is (D) 25x² - 5x - 2.