Zeros Of Polynomial Function: X³ + 9x² - 4x - 36

Determining the zeros of a polynomial function is a fundamental task in algebra, with applications spanning various fields of mathematics, science, and engineering. In this article, we will delve into the process of finding the zeros of the cubic function f(x) = x³ + 9x² - 4x - 36. We'll explore different techniques, including factoring by grouping and the rational root theorem, to systematically identify the values of x that make the function equal to zero. Understanding how to find these zeros is crucial for analyzing the behavior of the function, graphing it accurately, and solving related equations and inequalities. Let's embark on this mathematical journey together and unravel the solutions to this intriguing problem.

Understanding Zeros of a Function

Before we dive into the specifics of our cubic function, let's clarify what we mean by the zeros of a function. The zeros, also known as roots or x-intercepts, are the values of x for which the function f(x) equals zero. In graphical terms, these are the points where the graph of the function intersects the x-axis. Finding the zeros of a function is essential because they provide crucial information about the function's behavior, such as where it changes sign and where it reaches its minimum or maximum values. For polynomial functions, the zeros are particularly important as they help us factor the polynomial and understand its structure. There are several methods for finding zeros, including factoring, using the quadratic formula (for quadratic functions), and employing numerical methods for higher-degree polynomials. The technique we choose often depends on the specific form of the function and the tools we have available. In the case of our cubic function, we'll explore a combination of algebraic techniques to pinpoint its zeros accurately.

Factoring by Grouping: A Strategic Approach

When confronted with a polynomial like f(x) = x³ + 9x² - 4x - 36, a strategic approach is often the most efficient way to find its zeros. One such strategy is factoring by grouping, a technique that involves rearranging terms and factoring out common factors in pairs. Let's apply this method to our cubic function. First, we can group the first two terms and the last two terms: (x³ + 9x²) + (-4x - 36). Next, we look for common factors within each group. In the first group, x² is a common factor, and in the second group, -4 is a common factor. Factoring these out, we get x²(x + 9) - 4(x + 9). Notice that we now have a common factor of (x + 9) in both terms. We can factor this out, resulting in (x + 9)(x² - 4). The expression x² - 4 is a difference of squares, which can be further factored as (x + 2)(x - 2). Thus, our completely factored function is f(x) = (x + 9)(x + 2)(x - 2). This factored form makes it easy to identify the zeros of the function, as we'll see in the next section.

Identifying the Zeros from the Factored Form

Now that we have successfully factored the function f(x) = x³ + 9x² - 4x - 36 into the form f(x) = (x + 9)(x + 2)(x - 2), finding the zeros becomes a straightforward process. The zeros of a function are the values of x that make the function equal to zero. In the factored form, this occurs when any of the factors equal zero. So, we set each factor equal to zero and solve for x: x + 9 = 0, x + 2 = 0, and x - 2 = 0. Solving these equations gives us x = -9, x = -2, and x = 2. These are the zeros of the function f(x). This means that the graph of the function intersects the x-axis at the points (-9, 0), (-2, 0), and (2, 0). Knowing the zeros is incredibly useful for sketching the graph of the function and understanding its overall behavior. We can also use this information to solve related equations and inequalities. For instance, we can determine the intervals where the function is positive or negative by examining the sign of the function between and around the zeros.

Verifying the Zeros

After finding the zeros of a function, it's always a good practice to verify our results. This ensures that we haven't made any errors in our calculations and that our solutions are indeed correct. To verify the zeros of f(x) = x³ + 9x² - 4x - 36, which we found to be -9, -2, and 2, we can substitute each of these values back into the original function and check if the result is zero. Let's start with x = -9: f(-9) = (-9)³ + 9(-9)² - 4(-9) - 36 = -729 + 729 + 36 - 36 = 0. Next, let's check x = -2: f(-2) = (-2)³ + 9(-2)² - 4(-2) - 36 = -8 + 36 + 8 - 36 = 0. Finally, let's verify x = 2: f(2) = (2)³ + 9(2)² - 4(2) - 36 = 8 + 36 - 8 - 36 = 0. Since the function equals zero for each of these values, we can confidently conclude that -9, -2, and 2 are indeed the zeros of the function f(x) = x³ + 9x² - 4x - 36. This verification step provides us with assurance that our solution is accurate.

Conclusion

In this exploration, we successfully identified the zeros of the cubic function f(x) = x³ + 9x² - 4x - 36. By employing the technique of factoring by grouping, we transformed the function into its factored form, f(x) = (x + 9)(x + 2)(x - 2). From this form, we easily determined the zeros to be -9, -2, and 2. We then verified these zeros by substituting them back into the original function, confirming their validity. Finding the zeros of a function is a crucial skill in algebra and calculus, with applications in various mathematical and scientific contexts. It allows us to understand the behavior of the function, sketch its graph, and solve related problems. The methods we've used here, such as factoring by grouping, are valuable tools in any mathematician's or problem-solver's toolkit. Understanding these techniques empowers us to tackle more complex problems and gain deeper insights into the world of functions and equations. For further exploration of polynomial functions and their zeros, you might find valuable resources at websites like Khan Academy.