Constructing A Degree 3 Polynomial With Given Zeros And Leading Coefficient

In the realm of mathematics, particularly within algebra, the construction of polynomials based on specified characteristics is a fundamental skill. This article delves into the process of finding a polynomial of degree 3, given its zeros and leading coefficient. Specifically, we aim to construct a polynomial with zeros at -2, 7, and -7, and a leading coefficient of 1. This exploration not only reinforces the understanding of polynomial structure but also highlights the relationship between roots and factors of a polynomial. Understanding how to construct polynomials from their zeros is crucial in various mathematical applications, including curve fitting, solving equations, and understanding the behavior of functions. This article will provide a step-by-step guide to constructing such a polynomial, emphasizing the underlying principles and the algebraic manipulations involved. Let's embark on this mathematical journey to unravel the intricacies of polynomial construction.

Understanding the Fundamentals

Before we dive into the specific problem, it's crucial to grasp the foundational concepts that underpin polynomial construction. A polynomial is an expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents. The degree of a polynomial is the highest power of the variable in the expression. For instance, a degree 3 polynomial, also known as a cubic polynomial, will have the general form ax³ + bx² + cx + d, where a, b, c, and d are coefficients, and a is not zero. The leading coefficient is the coefficient of the term with the highest degree, which in this case is a. The zeros of a polynomial, also known as roots, are the values of the variable that make the polynomial equal to zero. These zeros are intimately connected to the factors of the polynomial. The Factor Theorem states that if r is a zero of a polynomial P(x), then (x - r) is a factor of P(x). This theorem is the cornerstone of constructing polynomials from their zeros. By understanding these fundamental concepts – the degree of a polynomial, the role of the leading coefficient, and the relationship between zeros and factors – we set the stage for constructing the desired polynomial with precision and clarity. The process involves translating zeros into factors, multiplying these factors together, and ensuring the leading coefficient aligns with the given value. This foundational knowledge not only aids in solving the current problem but also empowers one to tackle a broader range of polynomial-related challenges.

Constructing the Polynomial in Factored Form

Given the zeros -2, 7, and -7, we can directly apply the Factor Theorem to construct the factors of our desired polynomial. Recall that if r is a zero, then (x - r) is a factor. Therefore, the zero -2 corresponds to the factor (x - (-2)), which simplifies to (x + 2). Similarly, the zero 7 corresponds to the factor (x - 7), and the zero -7 corresponds to the factor (x - (-7)), which simplifies to (x + 7). Since we are looking for a degree 3 polynomial, and we have three zeros, we can express the polynomial P(x) in factored form as a product of these factors, multiplied by the leading coefficient. Given that the leading coefficient is 1, the polynomial in factored form is: P(x) = 1 * (x + 2) * (x - 7) * (x + 7). This factored form elegantly encapsulates the relationship between the zeros and the polynomial. Each factor directly corresponds to a zero, making it easy to verify the roots of the polynomial. The leading coefficient of 1 ensures that the polynomial's highest degree term will have a coefficient of 1 when the polynomial is expanded. This factored form is not only a valid representation of the polynomial but also a useful one for analyzing its behavior, such as identifying its zeros and understanding its graph. The next step involves expanding this factored form to obtain the polynomial in standard form, which will further illuminate its structure and properties. However, the factored form itself is a complete and concise answer to the problem.

Expanding the Factored Form to Standard Form

While the factored form of the polynomial, P(x) = (x + 2)(x - 7)(x + 7), provides a clear representation of its zeros, expanding it to the standard form ax³ + bx² + cx + d reveals additional insights into the polynomial's coefficients and overall structure. To expand the factored form, we multiply the factors together step by step. First, we can multiply the factors (x - 7) and (x + 7). Recognizing this as a difference of squares, (a - b)(a + b) = a² - b², we have (x - 7)(x + 7) = x² - 49. Now, we multiply this result by the remaining factor (x + 2): (x + 2)(x² - 49) = x(x² - 49) + 2(x² - 49). Distributing x and 2, we get: x³ - 49x + 2x² - 98. Rearranging the terms in descending order of exponents, we obtain the polynomial in standard form: P(x) = x³ + 2x² - 49x - 98. This standard form clearly shows the coefficients of each term: the coefficient of x³ is 1 (the leading coefficient, as specified), the coefficient of x² is 2, the coefficient of x is -49, and the constant term is -98. The standard form is useful for various purposes, such as evaluating the polynomial at specific values of x, analyzing its end behavior, and performing algebraic manipulations like polynomial division. Expanding the factored form to standard form provides a more comprehensive understanding of the polynomial's characteristics and facilitates its application in different mathematical contexts. This process highlights the flexibility of representing polynomials in different forms, each offering unique advantages depending on the task at hand.

Verifying the Zeros of the Polynomial

After constructing the polynomial, whether in factored form P(x) = (x + 2)(x - 7)(x + 7) or standard form P(x) = x³ + 2x² - 49x - 98, it is crucial to verify that the given zeros indeed satisfy the polynomial equation. This verification process not only confirms the correctness of our construction but also reinforces the understanding of the relationship between zeros and polynomial values. To verify the zeros, we substitute each zero into the polynomial and check if the result is zero. For the zero -2, we substitute x = -2 into the polynomial in standard form: P(-2) = (-2)³ + 2(-2)² - 49(-2) - 98 = -8 + 8 + 98 - 98 = 0. This confirms that -2 is indeed a zero of the polynomial. Next, we verify the zero 7: P(7) = (7)³ + 2(7)² - 49(7) - 98 = 343 + 98 - 343 - 98 = 0. This confirms that 7 is a zero as well. Finally, we verify the zero -7: P(-7) = (-7)³ + 2(-7)² - 49(-7) - 98 = -343 + 98 + 343 - 98 = 0. This confirms that -7 is also a zero. Since all three given zeros satisfy the polynomial equation, we can confidently conclude that our constructed polynomial is correct. This verification process highlights the importance of checking solutions in mathematical problems, ensuring accuracy and reinforcing the understanding of the underlying concepts. Furthermore, it demonstrates the consistency between the factored form and the standard form of the polynomial, as the zeros can be easily identified from the factored form and verified in the standard form.

Conclusion

In conclusion, we have successfully constructed a degree 3 polynomial with zeros -2, 7, and -7, and a leading coefficient of 1. We began by understanding the fundamental relationship between the zeros of a polynomial and its factors, utilizing the Factor Theorem. We then constructed the polynomial in factored form, P(x) = (x + 2)(x - 7)(x + 7), which directly reflects the given zeros. To gain further insight into the polynomial's structure, we expanded the factored form to obtain the standard form, P(x) = x³ + 2x² - 49x - 98. This standard form revealed the coefficients of each term, providing a more comprehensive representation of the polynomial. Crucially, we verified that the given zeros indeed satisfy the polynomial equation, ensuring the accuracy of our construction. This process of constructing and verifying polynomials from their zeros is a fundamental skill in algebra, with applications in various mathematical fields. It reinforces the understanding of polynomial structure, the relationship between roots and factors, and the importance of verification in mathematical problem-solving. By mastering these concepts, one can confidently tackle a wide range of polynomial-related challenges and appreciate the elegance and interconnectedness of mathematical ideas. The ability to move between factored and standard forms, and to verify solutions, is a testament to a deep understanding of polynomial algebra.