Finding Zeros Of Cubic Function G(x) = X³ + 6x² - 9x - 54

In this comprehensive article, we delve into the fascinating world of polynomial functions, specifically focusing on finding the zeros of a cubic function. Our main objective is to determine the zeros of the given cubic function g(x) = x³ + 6x² - 9x - 54. Understanding the zeros of a function is crucial in various fields of mathematics and its applications, as they represent the points where the function intersects the x-axis. This exploration will not only provide the solution to the problem but also enhance your understanding of polynomial functions and their properties. The main approach will be using factoring techniques to simplify the cubic equation into linear factors, subsequently revealing the roots or zeros of the function. We will also discuss the significance of these zeros in the context of the graph of the function, and explore other methods for finding zeros, such as the Rational Root Theorem. Understanding how to find the zeros of polynomials is a fundamental skill in algebra and calculus, with applications ranging from curve sketching to solving real-world problems involving rates of change and optimization. This article will act as a guide to systematically approach this kind of problems.

Before we dive into solving for the zeros of the cubic function g(x) = x³ + 6x² - 9x - 54, let's first establish a clear understanding of what zeros of a function are. Zeros, also known as roots or x-intercepts, are the values of x for which the function g(x) equals zero. In simpler terms, they are the points where the graph of the function intersects the x-axis. Finding these zeros is a fundamental task in algebra and calculus, as they provide key insights into the behavior of the function. For a polynomial function, the zeros determine where the function changes sign, which is critical for solving inequalities and understanding the function's graph. The number of zeros a polynomial has is directly related to its degree; a cubic function, like the one we are dealing with, can have up to three zeros, though some may be repeated or complex. These zeros are crucial in applications ranging from engineering to economics, where they help to model and predict outcomes in various systems. Understanding the concept of zeros and how to find them allows for a deeper analysis of the function's behavior and its practical implications. In this article, we will systematically find the zeros of the given cubic function, providing a detailed explanation of each step involved.

There are several methods to find the zeros of polynomials, each with its own advantages and applicability depending on the polynomial's characteristics. For linear and quadratic functions, straightforward algebraic methods like solving linear equations or using the quadratic formula are effective. However, for polynomials of higher degrees, such as our cubic function g(x) = x³ + 6x² - 9x - 54, the methods become more intricate. One common approach is factoring, where we attempt to express the polynomial as a product of simpler polynomials. This can be particularly useful if we can identify a rational root, which allows us to divide the polynomial and reduce its degree. Another method is the Rational Root Theorem, which provides a list of potential rational roots based on the polynomial's coefficients. This theorem is invaluable for narrowing down the possibilities when trying to find roots by trial and error. For more complex polynomials, numerical methods like the Newton-Raphson method can be employed to approximate the zeros. These methods are particularly useful when analytical solutions are difficult or impossible to find. In our case, we will primarily use factoring and the Rational Root Theorem to determine the zeros of the cubic function. Understanding these various methods equips us with a versatile toolkit for tackling different types of polynomial equations. By applying these techniques, we can systematically uncover the roots and gain a deeper insight into the function's behavior.

Now, let's apply the factoring method to find the zeros of the given cubic function, g(x) = x³ + 6x² - 9x - 54. Factoring is a powerful technique that simplifies the process of finding roots by breaking down the polynomial into simpler, more manageable factors. The first step in factoring is often to look for common factors among the terms. In this case, there isn't a single common factor across all terms, so we will proceed with grouping. Grouping involves pairing terms together and factoring out common factors from each pair. For our function, we can group the first two terms and the last two terms: (x³ + 6x²) + (-9x - 54). From the first group, we can factor out x², and from the second group, we can factor out -9. This gives us x²(x + 6) - 9(x + 6). Notice that we now have a common factor of (x + 6) in both terms. We can factor this out to get (x + 6)(x² - 9). The expression x² - 9 is a difference of squares, which can be further factored into (x + 3)(x - 3). Thus, our fully factored form of the cubic function is g(x) = (x + 6)(x + 3)(x - 3). To find the zeros, we set each factor equal to zero and solve for x. This gives us the equations x + 6 = 0, x + 3 = 0, and x - 3 = 0. Solving these equations yields the zeros x = -6, x = -3, and x = 3. This methodical approach to factoring allows us to break down a complex polynomial into its constituent linear factors, making it straightforward to determine the zeros.

After finding the zeros of the cubic function g(x) = x³ + 6x² - 9x - 54 using factoring, it is essential to verify these zeros to ensure the accuracy of our solution. Verification involves substituting each potential zero back into the original function and confirming that the result is indeed zero. This step is crucial because it helps to catch any errors made during the factoring process or while solving the linear equations. Our potential zeros are x = -6, x = -3, and x = 3. Let's substitute each value into g(x):

1. For x = -6: g(-6) = (-6)³ + 6(-6)² - 9(-6) - 54 = -216 + 216 + 54 - 54 = 0

2. For x = -3: g(-3) = (-3)³ + 6(-3)² - 9(-3) - 54 = -27 + 54 + 27 - 54 = 0

3. For x = 3: g(3) = (3)³ + 6(3)² - 9(3) - 54 = 27 + 54 - 27 - 54 = 0

As we can see, substituting each value into the function results in zero, confirming that x = -6, x = -3, and x = 3 are indeed the zeros of the cubic function g(x). This verification step not only confirms the correctness of our solution but also reinforces our understanding of the relationship between zeros and the function's value. It demonstrates that the points we found are precisely where the function's graph intersects the x-axis, providing a solid foundation for further analysis and applications.

Having successfully found and verified the zeros of the cubic function g(x) = x³ + 6x² - 9x - 54, we now have a clear understanding of the function's behavior. The zeros, which we determined to be x = -6, x = -3, and x = 3, represent the points where the function's graph intersects the x-axis. These zeros are critical in understanding the function's overall shape and behavior. By knowing the zeros, we can sketch the graph of the function more accurately, identify intervals where the function is positive or negative, and solve related inequalities. The zeros also provide valuable information about the function's factors, as each zero corresponds to a linear factor of the polynomial. In our case, the factors are (x + 6), (x + 3), and (x - 3), which give us a complete factorization of the cubic function. Looking at the given options, we can now select the correct answer based on our findings. The option that lists the zeros as 3, -3, -6 is the correct choice. This exercise highlights the importance of a systematic approach to solving mathematical problems, from factoring to verification, and emphasizes how each step contributes to a comprehensive understanding of the function. The accurate determination of zeros is a fundamental skill in mathematics, with far-reaching applications in various fields.

Based on our detailed analysis, the correct answer is:

B. 3, -3, -6

In conclusion, this article has provided a comprehensive guide to finding the zeros of the cubic function g(x) = x³ + 6x² - 9x - 54. We began by understanding the concept of zeros, their significance, and various methods to find them. We then applied the factoring method, a powerful technique for breaking down polynomials into simpler factors. By grouping terms and factoring out common factors, we successfully expressed the cubic function as a product of linear factors: g(x) = (x + 6)(x + 3)(x - 3). Setting each factor equal to zero, we determined the zeros to be x = -6, x = -3, and x = 3. To ensure the accuracy of our solution, we verified these zeros by substituting them back into the original function and confirming that the result was indeed zero. This step is crucial in mathematics to catch any potential errors and reinforce our understanding of the concepts. Finally, we analyzed the results and confidently selected the correct answer from the given options. This exercise highlights the importance of a systematic approach to problem-solving, from initial understanding to verification of results. The ability to find zeros of polynomial functions is a fundamental skill in mathematics, with applications in various fields, including engineering, physics, and economics. By mastering these techniques, one can gain a deeper understanding of functions and their behavior, enabling the solution of complex problems and the modeling of real-world phenomena.