Zeros Of Function Y=x(x-2)(x+6)^2: How To Find Them?

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\nLet's dive into finding the zeros of the function y = x(x-2)(x+6)^2. This might sound intimidating, but it's actually a straightforward process once you understand the basic concept. In simple terms, the zeros of a function are the values of 'x' that make 'y' equal to zero. These are also sometimes called roots or x-intercepts of the function. Finding these zeros is a fundamental skill in algebra and calculus, and it helps us understand the behavior of the function. This article will break down the process step-by-step, ensuring you grasp the concept fully. We'll not only identify the correct zeros but also explain why those values make the function equal to zero. So, grab your thinking cap, and let's get started on this mathematical adventure!

Understanding Zeros of a Function

When we talk about the zeros of a function, we're essentially asking: "For what values of 'x' does the function 'y' become zero?" In graphical terms, these are the points where the function's graph intersects the x-axis. To find these zeros, we set the function equal to zero and solve for 'x'. This is a crucial concept in algebra and calculus because it helps us understand the behavior of the function, such as where it crosses the x-axis, where it's positive or negative, and more. For polynomial functions, like the one we're dealing with y = x(x-2)(x+6)^2, each factor corresponds to a potential zero. The power of each factor tells us about the multiplicity of the zero, which affects how the graph behaves at that point. For instance, a factor raised to an even power (like the (x+6)^2 in our function) means the graph touches the x-axis at that point but doesn't cross it. Understanding these nuances is key to accurately sketching the graph and interpreting the function's properties. So, let's proceed with our specific function and identify its zeros.

Step-by-Step Solution

To find the zeros of the function y = x(x-2)(x+6)^2, we need to set the function equal to zero and solve for 'x'. This means we're looking for the values of 'x' that make the entire expression equal to zero. The beauty of this function is that it's already factored for us, which makes our job much easier.

Here’s how we do it:

  1. Set the function equal to zero:

    x(x-2)(x+6)^2 = 0

  2. Identify the factors:

    We have three factors here: x, (x-2), and (x+6)^2.

  3. Set each factor equal to zero:

    • x = 0
    • x - 2 = 0
    • (x+6)^2 = 0

  4. Solve each equation for x:

    • From x = 0, we directly get x = 0.
    • From x - 2 = 0, we add 2 to both sides to get x = 2.
    • From (x+6)^2 = 0, we take the square root of both sides to get x + 6 = 0, which gives us x = -6.

So, the zeros of the function are x = 0, x = 2, and x = -6. Notice that the factor (x+6)^2 has a multiplicity of 2, meaning that x = -6 is a repeated root.

Analyzing the Options

Now that we've found the zeros of the function y = x(x-2)(x+6)^2, let's compare our results with the given options to determine the correct answer. Our calculated zeros are x = -6, 0, and 2. We need to find the option that matches these values exactly.

  • Option A: -6, 0, and 2
  • Option B: -6, 0, 2, and 6
  • Option C: -2, 0, and 6
  • Option D: -6 and 2

By comparing our calculated zeros with the options, we can see that Option A perfectly matches our results. It includes all the zeros we found: -6, 0, and 2. Therefore, Option A is the correct answer. The other options either include incorrect values or miss some of the zeros, making them incorrect choices. Understanding how to find these zeros is crucial for analyzing polynomial functions and their graphs.

Why Option A is Correct

Option A, which states that the zeros of the function y = x(x-2)(x+6)^2 are -6, 0, and 2, is the correct answer. This is because when we substitute each of these values into the function, the function evaluates to zero. Let's verify this:

  • For x = -6:

    y = (-6)((-6)-2)((-6)+6)^2 = (-6)(-8)(0)^2 = 0

  • For x = 0:

    y = (0)((0)-2)((0)+6)^2 = (0)(-2)(6)^2 = 0

  • For x = 2:

    y = (2)((2)-2)((2)+6)^2 = (2)(0)(8)^2 = 0

As you can see, each value makes the function equal to zero, confirming that they are indeed the zeros of the function. It's important to understand that these zeros represent the points where the graph of the function intersects the x-axis. In the case of x = -6, it's a repeated root due to the (x+6)^2 term, meaning the graph touches the x-axis at this point but doesn't cross it. This nuanced understanding is crucial for accurately sketching and interpreting polynomial functions.

Common Mistakes to Avoid

When finding the zeros of a function, it's easy to make mistakes if you're not careful. Here are some common pitfalls to avoid:

  • Forgetting to set each factor to zero: Make sure you set each factor of the function equal to zero. For example, in the function y = x(x-2)(x+6)^2, you need to consider x = 0, x - 2 = 0, and (x+6)^2 = 0.
  • Incorrectly solving for x: Double-check your algebra when solving each equation. A simple mistake in arithmetic can lead to the wrong answer.
  • Ignoring multiplicity: Remember that if a factor is raised to a power (like (x+6)^2), it indicates the multiplicity of the root. This affects the behavior of the graph at that point.
  • Including extraneous solutions: Sometimes, when solving equations, you might end up with solutions that don't actually satisfy the original equation. Always check your solutions by plugging them back into the original function.
  • Misinterpreting the question: Make sure you understand what the question is asking. In this case, we're looking for the zeros of the function, which are the x-values that make the function equal to zero.

By being mindful of these common mistakes, you can increase your accuracy and confidence in finding the zeros of any function.

In conclusion, finding the zeros of a function involves setting the function equal to zero and solving for 'x.' For the function y = x(x-2)(x+6)^2, the zeros are -6, 0, and 2. Therefore, the correct answer is A. Understanding how to find and interpret these zeros is essential for analyzing the behavior of functions in mathematics. For further exploration, you might find valuable insights on function analysis at Khan Academy.