X-Intercepts: F(x) = X(x+11)^2(x-4)

Hey guys! Today, we're diving into the fascinating world of polynomial functions, specifically focusing on how to find their x-intercepts. X-intercepts, also known as roots or zeros, are the points where the graph of the function crosses or touches the x-axis. These points are crucial for understanding the behavior of the function and sketching its graph. We'll break down the process step-by-step, ensuring you grasp the concept and can confidently tackle any polynomial function. So, let's get started and unravel the mystery behind x-intercepts!

Understanding X-Intercepts

Before we jump into solving, let's clarify what x-intercepts really are. The x-intercept is the point where the graph of a function intersects the x-axis. At this point, the y-value (or f(x) value) is always zero. Therefore, to find the x-intercepts, we set f(x) = 0 and solve for x. This is a fundamental concept in algebra and calculus, and mastering it will significantly improve your problem-solving skills. Remember, each x-intercept corresponds to a real root of the polynomial equation. The number of x-intercepts a polynomial function can have is at most equal to its degree. For instance, a cubic polynomial (degree 3) can have up to 3 x-intercepts. Understanding this connection between x-intercepts and roots helps in visualizing the graph of the polynomial and predicting its behavior. Moreover, identifying x-intercepts is often the first step in analyzing and graphing polynomial functions, providing a crucial foundation for further analysis. So, keep this definition in mind as we move forward!

Steps to Find X-Intercepts

Finding the x-intercepts of a polynomial function involves a few key steps. First, set the function equal to zero. This is because, at the x-intercept, the value of f(x) is zero. This transforms the problem into solving an equation. Next, factor the polynomial if possible. Factoring helps break down the polynomial into simpler expressions, making it easier to find the values of x that make the function zero. If the polynomial is not easily factorable, you might need to use techniques like synthetic division or the rational root theorem. Once you have factored the polynomial, set each factor equal to zero. This is based on the zero-product property, which states that if the product of several factors is zero, then at least one of the factors must be zero. Solving each of these equations will give you the x-values of the x-intercepts. Finally, write the x-intercepts as coordinate pairs (x, 0). Remember, the y-coordinate of an x-intercept is always zero. By following these steps, you can systematically find all the x-intercepts of a given polynomial function. It's a straightforward process, but it requires careful attention to detail and a good understanding of factoring techniques.

Example: f(x) = x(x+11)^2(x-4)

Let's apply these steps to the polynomial function given: f(x) = x(x+11)^2(x-4). The first step is to set f(x) equal to zero: 0 = x(x+11)^2(x-4). Notice that this polynomial is already conveniently factored for us, which saves a lot of time and effort. Now, we set each factor equal to zero: x = 0, (x+11)^2 = 0, and (x-4) = 0. Solving each of these equations gives us the x-values of the x-intercepts. For x = 0, we have one x-intercept at x = 0. For (x+11)^2 = 0, we take the square root of both sides to get x+11 = 0, which gives us x = -11. Note that the factor (x+11) is squared, meaning that x = -11 is a root with multiplicity 2. This tells us that the graph of the function touches the x-axis at x = -11 but does not cross it. For (x-4) = 0, we add 4 to both sides to get x = 4. Therefore, the x-intercepts are x = 0, x = -11, and x = 4. Writing these as coordinate pairs, we have (0, 0), (-11, 0), and (4, 0). These are the points where the graph of the function intersects or touches the x-axis. In summary, by setting the function to zero, factoring (or recognizing the factored form), and solving for x, we successfully found all the x-intercepts of the given polynomial function.

Analyzing the Options

Now, let's analyze the given options to determine which one correctly identifies all the x-intercepts of the polynomial function f(x) = x(x+11)^2(x-4). Option A states the x-intercepts as (0,0), (11,0), and (-4,0). We found the x-intercepts to be at x=0, x=-11 and x=4. Thus, this option is incorrect because it incorrectly identifies x = 11 and x = -4 as x-intercepts. Option B states the x-intercepts as (0,0), (-11,0), and (4,0). Comparing this with our solution, we see that this option correctly identifies all three x-intercepts: x = 0, x = -11, and x = 4. Therefore, option B is the correct answer. Option C states the x-intercepts as (0,11) and (0,4). This option is incorrect because it confuses the x and y coordinates and does not include the x-intercept at (0, 0). Option D states the x-intercepts as (-11,0) and (4,0). While this option correctly identifies the x-intercepts at x = -11 and x = 4, it is incomplete because it omits the x-intercept at x = 0. Therefore, option D is also incorrect. By carefully comparing each option with our calculated x-intercepts, we can confidently determine that only option B accurately represents all the x-intercepts of the given polynomial function.

The Correct Answer

After carefully analyzing the options and solving for the x-intercepts, we can confidently conclude that the correct answer is:

B. (0,0), (-11,0), (4,0)

This option accurately identifies all the points where the polynomial function f(x) = x(x+11)^2(x-4) intersects the x-axis. Remember, finding x-intercepts is a crucial skill in understanding and graphing polynomial functions. Keep practicing, and you'll become a pro in no time! Understanding x-intercepts helps in graphing polynomial functions and solving related problems. Happy problem-solving, guys! Keep up the great work!