Second Degree Polynomial Function With Leading Coefficient 2 And Roots -3 And 5

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In this article, we will delve into the fascinating world of polynomial functions, specifically focusing on second-degree polynomials, also known as quadratic functions. We'll explore how to construct such a function given its leading coefficient and roots. This is a fundamental concept in algebra and has wide applications in various fields, including physics, engineering, and economics. Understanding the relationship between the roots, leading coefficient, and the quadratic function itself is crucial for solving many mathematical problems. We will dissect the problem step by step, providing a clear and concise explanation of the underlying principles and calculations. By the end of this discussion, you will have a solid grasp of how to determine a quadratic function based on its key characteristics, setting you up for success in your mathematical endeavors. So, let's embark on this journey to unlock the secrets of quadratic functions and their roots.

At its core, a quadratic function is a polynomial function of degree two. This means that the highest power of the variable in the function is 2. The general form of a quadratic function is expressed as f(x) = ax^2 + bx + c, where a, b, and c are constants, and a is not equal to zero. The coefficient a is particularly important as it determines the leading coefficient of the quadratic function, which plays a significant role in shaping the parabola, the graphical representation of a quadratic function. The roots of a quadratic function, also known as the zeros or solutions, are the values of x for which f(x) = 0. These roots are the points where the parabola intersects the x-axis. A quadratic function can have two distinct real roots, one real root (a repeated root), or no real roots. The roots are intrinsically linked to the coefficients of the quadratic function, and understanding this relationship is crucial for solving quadratic equations and constructing quadratic functions. In this context, we will explore how the leading coefficient and the roots dictate the specific form of the quadratic function, allowing us to identify the correct function from a set of options.

Our task is to identify the second-degree polynomial function that satisfies two specific conditions: a leading coefficient of 2 and roots of -3 and 5. In mathematical terms, we are looking for a quadratic function f(x) = ax^2 + bx + c where a = 2 and the function equals zero when x = -3 and x = 5. This problem requires us to reverse-engineer the quadratic function from its roots and leading coefficient. We know that the roots of a quadratic function are directly related to its factored form. If r1 and r2 are the roots of a quadratic function, then the function can be written in the form f(x) = a(x - r1)(x - r2), where a is the leading coefficient. In our case, r1 = -3, r2 = 5, and a = 2. By substituting these values into the factored form and expanding the expression, we can determine the quadratic function that satisfies the given conditions. This process will involve careful algebraic manipulation to ensure we arrive at the correct coefficients for the x^2, x, and constant terms. The problem highlights the interplay between the roots of a polynomial and its algebraic representation, a fundamental concept in polynomial theory.

To solve this problem, we will employ the factored form of a quadratic function. As mentioned earlier, if a quadratic function has roots r1 and r2 and a leading coefficient a, it can be expressed as f(x) = a(x - r1)(x - r2). This form is particularly useful when the roots are known because it directly incorporates them into the function's expression. In our case, we have a = 2, r1 = -3, and r2 = 5. Substituting these values into the factored form, we get f(x) = 2(x - (-3))(x - 5), which simplifies to f(x) = 2(x + 3)(x - 5). The next step is to expand this expression to obtain the quadratic function in the standard form f(x) = ax^2 + bx + c. Expanding the expression involves multiplying the two binomials (x + 3) and (x - 5) and then multiplying the resulting expression by 2. This process will give us the coefficients of the quadratic function, allowing us to identify the correct option from the given choices. This approach leverages the fundamental relationship between the roots and the factored form of a quadratic function, providing a systematic way to construct the function.

Let's meticulously walk through the solution process:

  1. Start with the factored form: We know that f(x) = a(x - r1)(x - r2), where a = 2, r1 = -3, and r2 = 5. Substituting these values, we get:

    f(x) = 2(x - (-3))(x - 5) f(x) = 2(x + 3)(x - 5)

  2. Expand the binomials: Multiply the two binomials (x + 3) and (x - 5):

    (x + 3)(x - 5) = x(x - 5) + 3(x - 5)

    •    = x^2 - 5x + 3x - 15*

    •    = x^2 - 2x - 15*

  3. Multiply by the leading coefficient: Multiply the resulting expression by the leading coefficient, which is 2:

    f(x) = 2(x^2 - 2x - 15) f(x) = 2x^2 - 4x - 30

  4. Identify the correct option: Comparing our result, f(x) = 2x^2 - 4x - 30, with the given options, we find that it matches option C.

Therefore, the second-degree polynomial function with a leading coefficient of 2 and roots -3 and 5 is f(x) = 2x^2 - 4x - 30.

Now, let's analyze the given options to understand why the other options are incorrect:

  • A. f(x) = 2x^2 + 4x - 30: This option has the correct leading coefficient (2) and constant term (-30), but the coefficient of the x term is incorrect. The correct coefficient should be -4, not +4. This indicates that the roots used to construct this function are not -3 and 5. If we were to find the roots of this function, they would be different from the specified roots.

  • B. f(x) = 2x^2 + 2x - 15: This option has the correct leading coefficient (2), but both the coefficient of the x term and the constant term are incorrect. The correct coefficients should be -4 and -30, respectively. This discrepancy indicates that the roots used to construct this function are significantly different from -3 and 5. The constant term, in particular, is half of what it should be, suggesting a fundamental error in the function's construction.

  • C. f(x) = 2x^2 - 4x - 30: This is the correct option, as we derived in our step-by-step solution. It has the correct leading coefficient (2), the correct coefficient of the x term (-4), and the correct constant term (-30). This function satisfies the given conditions of having roots -3 and 5 and a leading coefficient of 2.

  • D. f(x) = 2x^2 - 2x - 15: This option has the correct leading coefficient (2), but both the coefficient of the x term and the constant term are incorrect. The correct coefficients should be -4 and -30, respectively. This suggests that the roots of this function are different from -3 and 5, and the function does not satisfy the given conditions.

By analyzing each option, we can clearly see why option C is the only one that satisfies the given conditions. The other options have incorrect coefficients, indicating that they do not have the specified roots or were constructed incorrectly.

In conclusion, the second-degree polynomial function with a leading coefficient of 2 and roots -3 and 5 is f(x) = 2x^2 - 4x - 30 (Option C). This was determined by utilizing the factored form of a quadratic function, substituting the given roots and leading coefficient, and then expanding the expression to obtain the standard form of the quadratic function. This problem highlights the fundamental relationship between the roots of a polynomial and its algebraic representation. The factored form provides a direct link between the roots and the function, making it a powerful tool for constructing polynomials with specific properties. By understanding this relationship, we can solve a wide range of problems involving polynomial functions. The step-by-step solution presented here demonstrates a systematic approach to solving such problems, ensuring accuracy and clarity. The analysis of the incorrect options further reinforces the understanding of why the correct answer is the only one that satisfies the given conditions. This problem serves as a valuable exercise in mastering the concepts of quadratic functions and their roots, essential for success in algebra and beyond.