Finding Zeros Of Function F(x) = X(x-1)(x+11) / (x+12)(x-13)

Hey guys! Today, we're diving deep into the fascinating world of functions, specifically to pinpoint the zeros of the function F(x) = x(x-1)(x+11) / (x+12)(x-13). Finding zeros might sound like a daunting task, but trust me, it's like piecing together a puzzle. Once you grasp the core concept, you'll be solving these problems like a pro. So, let's break it down and make sure we understand each step along the way. We're not just solving for an answer here; we're building a solid foundation in understanding functions. Zeros, in the simplest terms, are the x-values that make the function equal to zero. They are also known as roots or x-intercepts, and they play a crucial role in understanding the behavior of a function. So, without further ado, let's get started and unlock the secrets behind those zeros!

What are Zeros of a Function?

Okay, before we jump into the specifics of our function, let's zoom out and get crystal clear on what zeros of a function actually are. Imagine you're looking at the graph of a function – maybe it's a wavy line, a curve, or something else entirely. The zeros of the function are precisely the points where that graph crosses or touches the x-axis. Think of the x-axis as the 'ground level' for our function's graph, and the zeros are where the function 'lands' on that ground. Mathematically speaking, these are the x-values that, when plugged into the function, make the whole thing equal to zero. So, we're essentially looking for the inputs (x-values) that produce an output of zero. Why are these zeros so important? Well, they give us key insights into the function's behavior. They help us understand where the function changes its sign (from positive to negative or vice versa), and they're crucial in many real-world applications, from engineering to economics. For instance, in physics, the zeros of a function might represent the points where a projectile hits the ground. In economics, they could represent the break-even points for a business. So, zeros aren't just abstract mathematical concepts; they're powerful tools for understanding and modeling the world around us. Let's keep this definition in mind as we tackle our function. We're not just hunting for numbers; we're uncovering the fundamental characteristics of the function itself. With a solid grasp of what zeros represent, we're well-equipped to tackle any function that comes our way. Remember, understanding the 'why' behind the math makes the 'how' much easier.

Finding the Zeros of F(x) = x(x-1)(x+11) / (x+12)(x-13)

Alright, let's get down to the nitty-gritty and find the zeros of our function: F(x) = x(x-1)(x+11) / (x+12)(x-13). Now, when we're dealing with a rational function like this – that is, a function that's expressed as a fraction where the numerator and denominator are polynomials – the key thing to remember is that a fraction is equal to zero only when its numerator is equal to zero. Think about it: if you have 0 divided by anything (except 0), the result is always 0. So, our mission here is to figure out what values of x will make the numerator of our function equal to zero. The numerator of F(x) is x(x-1)(x+11). This is where things get interesting because we have a product of three factors: x, (x-1), and (x+11). For this entire product to be zero, at least one of these factors must be zero. This is a crucial concept known as the zero-product property. It's like a golden rule in algebra, and it simplifies our task immensely. So, let's take each factor and set it equal to zero, one by one. First, we have x = 0. That's straightforward enough – one of our zeros is x = 0. Next up is (x-1) = 0. To solve for x, we simply add 1 to both sides, giving us x = 1. So, there's another zero! Finally, we have (x+11) = 0. Subtracting 11 from both sides, we find x = -11. And there we have it – three potential zeros: 0, 1, and -11. But hold on a second! We're not quite done yet. Remember, we have a denominator in our function as well: (x+12)(x-13). We need to make sure that these values don't make the denominator equal to zero, because division by zero is a big no-no in mathematics (it's undefined). So, let's quickly check. If x = 0, the denominator is (12)(-13), which is not zero. If x = 1, the denominator is (13)(-12), again, not zero. And if x = -11, the denominator is (1)(-24), still not zero. Phew! So, our three zeros – 0, 1, and -11 – are indeed valid. They make the numerator zero without making the denominator zero. This is a critical step in finding zeros of rational functions, as we need to exclude any values that would lead to division by zero. With that final check, we can confidently say that the zeros of F(x) are 0, 1, and -11.

Why the Denominator Matters

Okay, we've successfully found the zeros of our function, but it's super important to really grasp why we had to check the denominator. It's not just some random step we threw in there; it's a fundamental part of working with rational functions. Remember, a rational function is essentially a fraction, and fractions have a golden rule: the denominator can never be zero. It's like a mathematical law of nature. Why is that? Well, division by zero is undefined in mathematics. It breaks the whole system, leading to nonsensical results. Think about it this way: division is the inverse of multiplication. When we say 10 / 2 = 5, we mean that 2 * 5 = 10. But what if we try to divide by zero? Let's say 10 / 0 = ?. What number, when multiplied by 0, gives us 10? There isn't one! Any number multiplied by 0 is 0, not 10. That's why division by zero is a mathematical black hole – it just doesn't work. Now, back to our function F(x) = x(x-1)(x+11) / (x+12)(x-13). We found the values that make the numerator zero, which are our potential zeros of the function. But if any of these values also make the denominator zero, then the function is undefined at that point. These values are called discontinuities or vertical asymptotes, and they're crucial for understanding the function's behavior. They represent points where the function 'blows up' to infinity or negative infinity, and they're not part of the function's zeros. So, when we checked the denominator, we were essentially making sure that our potential zeros weren't actually points of discontinuity. It's like double-checking your map before you embark on a journey – you want to make sure you're heading in the right direction and not towards a cliff! In our case, the values that make the denominator zero are x = -12 and x = 13. If we had found that any of our potential zeros were -12 or 13, we would have had to exclude them from our list of actual zeros. They're important points on the graph, but they're not where the function equals zero. They're where the function becomes undefined. So, the denominator plays a critical role in defining the function's domain – the set of all possible input values. By understanding the denominator, we gain a deeper insight into the function's behavior and its overall graph. It's not just about finding zeros; it's about understanding the complete picture.

Choosing the Correct Answer

Alright, now that we've done the detective work and found the zeros of our function, let's circle back to the original question and pinpoint the correct answer. Remember, we were given the function F(x) = x(x-1)(x+11) / (x+12)(x-13), and we were tasked with finding its zeros. Through our step-by-step process, we carefully analyzed the function, focusing on when the numerator equals zero while ensuring the denominator doesn't also become zero. We found three distinct values that satisfy this condition: x = 0, x = 1, and x = -11. These are the heroes of our story – the x-values that make the function's output zero. Now, let's look at the answer choices provided. We need to find the one that lists exactly these values: 0, 1, and -11. Option A presents us with 0, 1, and -11. Bingo! That's the one we've been searching for. It perfectly matches the zeros we calculated. Option B lists -13 and 12. These are actually the values that make the denominator zero, not the numerator. They're the troublemakers that make the function undefined, not the zeros we're after. Option C gives us 0, -1, and 11. While 0 is indeed a zero, -1 and 11 are not. We found that the correct values should be 1 and -11, not their opposites. Option D offers 13 and -12. Just like in Option B, these are the values that make the denominator zero, not the numerator. They're the points of discontinuity, not the zeros. So, after carefully comparing our findings with the answer choices, it's crystal clear that Option A – 0, 1, and -11 – is the correct answer. We've not only solved the problem but also understood why this is the correct answer. We've delved into the mechanics of finding zeros, the importance of the denominator, and the process of verifying our solution against the given options. This isn't just about getting the right answer; it's about building a solid understanding of the underlying concepts.

Key Takeaways

Okay, guys, we've reached the end of our journey into the zeros of functions, and it's time to recap the key takeaways from our exploration. We didn't just find the answer; we uncovered the fundamental principles behind it. So, what are the big ideas we should remember? First and foremost, we learned that the zeros of a function are the x-values that make the function equal to zero. They're the points where the graph of the function intersects or touches the x-axis. Think of them as the 'ground level' points for the function. Understanding this definition is the cornerstone of finding zeros. Next, we focused on how to find zeros, especially for rational functions like our example, F(x) = x(x-1)(x+11) / (x+12)(x-13). The golden rule here is: set the numerator equal to zero and solve for x. This is because a fraction is zero only when its numerator is zero (as long as the denominator isn't also zero). We used 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. This allowed us to break down the numerator into simpler equations and find the potential zeros. But here's the crucial twist: we must always check the denominator! This is where many students can stumble, but we're not going to let that happen to us. The denominator can never be zero, so we need to make sure that none of our potential zeros also make the denominator zero. If they do, they're not zeros at all; they're points of discontinuity or vertical asymptotes. They're important features of the function, but they don't belong on the list of zeros. Finally, we emphasized the importance of understanding why we do each step. It's not enough to just memorize a procedure; we need to grasp the underlying concepts. This deeper understanding allows us to tackle a wider range of problems and apply our knowledge in different contexts. So, remember, zeros are fundamental to understanding the behavior of a function. They tell us where the function crosses the x-axis, and they're crucial in many real-world applications. By setting the numerator to zero, checking the denominator, and understanding the 'why' behind the math, we can confidently find the zeros of any rational function that comes our way. Keep practicing, keep exploring, and keep unlocking the secrets of mathematics!

This detailed explanation should give anyone a solid understanding of how to find the zeros of a rational function and why each step is important.