Understanding Function Zeros: A Deep Dive

Hey there, math enthusiasts! Today, we're diving deep into the fascinating world of function zeros. Ever wondered what those 'zeros' actually represent? They're essentially the points where a function crosses the x-axis, the very values of 'x' that make the function's output equal to zero. Understanding these points is crucial for graphing functions, solving equations, and unlocking a deeper understanding of mathematical relationships. We'll be exploring a specific function, $h(x)=(x-6)(x^2+8x+16)$, and unraveling the nature of its zeros. This exploration will not only help us answer a specific question about this function but also equip us with the tools to analyze any function's zeros. So, grab your thinking caps, and let's get started on this mathematical journey! We'll break down the function, identify its components, and systematically determine the number and type of its zeros. This process involves a bit of algebraic manipulation and a clear understanding of quadratic expressions, but don't worry, we'll walk through it step-by-step. The goal is to move beyond simply finding answers and to truly grasp the 'why' behind them, fostering a more robust mathematical intuition. Get ready to see how different parts of a function contribute to its overall behavior and how seemingly complex expressions can be simplified to reveal their fundamental properties.

Deconstructing the Function: $h(x)=(x-6)(x^2+8x+16)$

Let's start by dissecting the function $h(x)=(x-6)(x^2+8x+16)$. This function is presented in a factored form, which is a huge advantage when we're looking for its zeros. The zeros of a function are the values of $x$ for which $h(x) = 0$. So, we need to find the values of $x$ that make the entire expression equal to zero. Because the function is a product of two parts, $(x-6)$ and $(x^2+8x+16)$, the entire expression will be zero if either of these parts is equal to zero. This is a fundamental property of multiplication: if any factor in a product is zero, the entire product becomes zero. This principle forms the bedrock of our approach to finding the zeros. We will set each factor equal to zero independently and solve for $x$. This strategy works because any value of $x$ that makes $(x-6)$ zero will make $h(x)$ zero, and similarly, any value of $x$ that makes $(x^2+8x+16)$ zero will also make $h(x)$ zero. Therefore, our task boils down to solving two separate equations: $x-6=0$ and $x^2+8x+16=0$. Each of these equations will yield potential zeros for our function $h(x)$.

Solving for the First Factor: $x-6=0$

Our first factor is the simple linear expression $x-6$. To find the value of $x$ that makes this factor zero, we set up the equation: $x-6 = 0$. This is a straightforward linear equation. To isolate $x$, we simply add 6 to both sides of the equation. This gives us $x = 6$. So, $x=6$ is one of the zeros of the function $h(x)$. This is a real zero because 6 is a real number. It's also a distinct zero, meaning it's a unique value that makes the function equal to zero. This part of the analysis is relatively simple, but it's crucial to remember that this is just one piece of the puzzle. The other part of the function, the quadratic expression, might introduce more zeros, or it might not. The nature of the zeros from the quadratic factor will significantly impact the total count and type of zeros for the entire function $h(x)$. Keep in mind that linear factors like $(x-6)$ always contribute exactly one real zero.

Tackling the Second Factor: $x^2+8x+16=0$

The second factor is a quadratic expression: $x^2+8x+16$. To find its zeros, we need to solve the quadratic equation $x^2+8x+16=0$. Now, quadratic equations can be solved in a few ways: factoring, completing the square, or using the quadratic formula. Let's see if this quadratic is easily factorable. We are looking for two numbers that multiply to 16 (the constant term) and add up to 8 (the coefficient of the $x$ term). The numbers 4 and 4 fit this description perfectly, as $4 imes 4 = 16$ and $4 + 4 = 8$. This means we can factor the quadratic expression as $(x+4)(x+4)$, or more compactly, as $(x+4)^2$. So, the equation becomes $(x+4)^2 = 0$. To solve for $x$, we take the square root of both sides, which gives us $x+4 = 0$. Subtracting 4 from both sides, we get $x = -4$. This is another real zero of the function $h(x)$.

The Nature of Repeated Zeros

Here's where things get interesting. We found that the quadratic factor $x^2+8x+16$ factors into $(x+4)^2$. This means that the value $x=-4$ is a repeated zero, also known as a zero with multiplicity 2. In essence, the factor $(x+4)$ appears twice. When we talk about distinct zeros, we mean unique values. So, while $x=-4$ arises from a factor that appears twice, it is still only one distinct value. If we were to list all the zeros without considering multiplicity, we would have 6 and -4. If we were to list them with multiplicity, we would have 6, -4, and -4. The question asks about distinct real zeros. This distinction is crucial for correctly interpreting the options. A zero with multiplicity 2 means that the graph of the function touches the x-axis at that point and then turns back, rather than crossing it. This behavior is characteristic of even multiplicities. For odd multiplicities, the graph crosses the x-axis.

Analyzing the Options

Now that we've found the zeros, let's revisit the options provided: A. The function has two distinct real zeros. B. The function has three distinct real zeros. C. The function has one real zero and... (the option is incomplete, but we can infer the analysis). We found two distinct real zeros: $x=6$ and $x=-4$. The zero $x=-4$ has a multiplicity of 2, but it is still only one distinct value. Therefore, the statement that best describes the zeros of the function $h(x)=(x-6)(x^2+8x+16)$ is that it has two distinct real zeros. This aligns with option A. It's important to differentiate between the total number of zeros (counting multiplicity) and the number of distinct zeros. In this case, counting multiplicity, we have three zeros (6, -4, -4), but only two unique values (6 and -4).

Conclusion: Embracing the Nuances of Function Zeros

Our exploration into the zeros of $h(x)=(x-6)(x^2+8x+16)$ has revealed a fundamental aspect of function analysis: the importance of distinguishing between distinct zeros and zeros counted with their multiplicity. We successfully identified that the function has two distinct real zeros: $x=6$ and $x=-4$. The zero $x=-4$ is a repeated zero, meaning it has a multiplicity of 2, stemming from the squared factor $(x+4)^2$. This multiplicity affects the behavior of the graph at $x=-4$, where it touches the x-axis and turns. Understanding these concepts is not just about solving textbook problems; it's about building a robust foundation for comprehending more complex mathematical models and real-world applications. Whether you're analyzing data, modeling physical phenomena, or delving into advanced calculus, the ability to accurately determine and interpret function zeros is an invaluable skill. Keep practicing, keep questioning, and remember that every mathematical concept, no matter how simple it may seem at first, holds a deeper layer of understanding waiting to be uncovered. For further exploration into the nuances of polynomial functions and their roots, you might find the resources at Khan Academy extremely helpful. Their comprehensive guides and interactive exercises can deepen your understanding of topics like factoring, quadratic equations, and the relationship between zeros and graphs.