Zeros Of Function G(s): Step-by-Step Solution

Hey guys! Ever found yourself staring blankly at a function, wondering where to even begin finding its zeros? Don't worry, we've all been there. Today, we're going to break down a specific example step-by-step, making the process super clear and (dare I say?) even a little fun. We're tackling the function g(s) = K(s + 2)(s + 10) / s(s + 1)(s + 5)(s + 15)². This might look intimidating at first, but trust me, we'll conquer it together. Let's dive into finding the zeros of this function. Remember, the zeros of a function are simply the values of 's' that make the function equal to zero. This is a fundamental concept in mathematics, and mastering it opens doors to understanding more complex topics. So, buckle up, and let’s get started!

Understanding Zeros of a Function

Before we jump into the specifics of our function, let's make sure we're all on the same page about what zeros actually are. In the simplest terms, the zeros of a function are the points where the function crosses the x-axis (or in this case, the s-axis). At these points, the value of the function is zero. Think of it like this: if you were to graph the function, the zeros would be where the line intersects the horizontal axis. Mathematically, a zero 's' of a function g(s) satisfies the equation g(s) = 0. This core principle is what guides our entire process. Now, why are zeros so important? Well, they tell us a lot about the behavior of the function. They help us understand where the function is positive, negative, and where it changes direction. In many real-world applications, zeros represent significant points, like equilibrium states in physics or break-even points in economics. So, understanding how to find them is a crucial skill in various fields. When dealing with rational functions like the one we have, zeros are particularly linked to the numerator. We'll see exactly why in the next section.

Identifying Potential Zeros from the Numerator

Okay, so we know that zeros make the function equal to zero. But how do we actually find them? For rational functions like our g(s), the key lies in the numerator. Remember, a fraction is equal to zero only if its numerator is zero (and the denominator is not). This is a critical concept! So, let's take a close look at the numerator of our function: K(s + 2)(s + 10). Notice the constant 'K'? Well, for the sake of finding zeros, we can essentially ignore it (unless K = 0, which would make the whole function zero, a rather boring case!). What we really care about are the factors (s + 2) and (s + 10). Each of these factors can potentially contribute a zero. Why? Because if either of these factors equals zero, the entire numerator becomes zero, and thus the function g(s) becomes zero. To find these potential zeros, we simply set each factor equal to zero and solve for 's'. This is a straightforward algebraic step that gets us closer to our solution. So, we'll have two simple equations to solve: (s + 2) = 0 and (s + 10) = 0. Solving these will reveal the values of 's' that make the numerator zero, giving us our candidate zeros.

Solving for the Zeros

Alright, let's get our hands dirty with some algebra! We've identified the factors in the numerator that could give us zeros: (s + 2) and (s + 10). Now, we'll set each of these equal to zero and solve for 's'. This is where the magic happens! First, let's tackle (s + 2) = 0. To isolate 's', we simply subtract 2 from both sides of the equation. This gives us s = -2. Ta-da! We've found one potential zero. Now, let's move on to the second factor: (s + 10) = 0. Similarly, we subtract 10 from both sides to isolate 's'. This results in s = -10. And there you have it, another potential zero! So, based on the numerator, we have two candidates for zeros: s = -2 and s = -10. But hold on, we're not quite done yet. We need to make sure these values don't also make the denominator zero, which would make the function undefined, not zero. This is a crucial step in finding zeros of rational functions. So, let's investigate the denominator in the next section.

Verifying Zeros by Checking the Denominator

We've found our potential zeros, s = -2 and s = -10, by looking at the numerator. Awesome! But before we declare victory, we need to do a vital check: ensure that these values don't also make the denominator equal to zero. Why? Because if the denominator is zero, the function is undefined at that point, not zero. This is a critical distinction! Let's take a look at our denominator: s(s + 1)(s + 5)(s + 15)². We need to see if plugging in s = -2 or s = -10 makes this expression equal to zero. If it does, then that value is not a zero of the function, but rather a point of discontinuity (like a vertical asymptote). So, let's start with s = -2. Plugging this into the denominator, we get: (-2)(-2 + 1)(-2 + 5)(-2 + 15)² = (-2)(-1)(3)(13)² This is clearly not zero. Phew! Now, let's try s = -10. Plugging this in, we get: (-10)(-10 + 1)(-10 + 5)(-10 + 15)² = (-10)(-9)(-5)(5)² Again, this is not zero. So, both s = -2 and s = -10 do not make the denominator zero. This means they are indeed the zeros of our function g(s). We've successfully verified our solutions! Let's wrap things up in the conclusion.

Conclusion: The Zeros of g(s)

Alright, guys, we did it! We successfully navigated the function g(s) = K(s + 2)(s + 10) / s(s + 1)(s + 5)(s + 15)² and found its zeros. We started by understanding the fundamental concept of zeros, then identified potential zeros from the numerator, and finally, we verified that these values didn't make the denominator zero. Through this process, we confidently arrived at our answer: the zeros of the function g(s) are s = -2 and s = -10. This might seem like a lot of steps, but each one is crucial to ensuring we find the correct solution. Remember, finding zeros is a key skill in mathematics and has applications in various fields. So, keep practicing, and you'll become a pro in no time! Hopefully, this step-by-step guide has made the process clear and easy to understand. Keep exploring those functions and uncovering their secrets!