Finding Zeros Of Polynomials How To Solve X(x – 3)(x + 5)(x² + 4) = 0

Hey guys! Ever stumbled upon an expression in math and wondered how to make it equal zero? Well, today, we’re diving deep into one such expression: x(x – 3)(x + 5)(x² + 4). Our mission? To find out which values of x will make this whole thing zero. We've got some options on the table: -5, 0, 3, and 4. Let’s break it down step by step, making sure we understand exactly why some of these work and others don't. Get ready to put on your math hats; it's going to be an insightful journey!

Understanding the Zero Product Property

Before we jump into the specifics, let's quickly recap a fundamental concept that’s going to be our guiding star here: the Zero Product Property. This property is the backbone of solving equations like ours. So, what does it say? Simply put, if you multiply a bunch of things together and the result is zero, then at least one of those things must be zero. Think of it like this: if you have a group of friends, and their combined effort results in nothing, then at least one person in the group isn't pulling their weight, right? Okay, maybe that's a bit harsh, but you get the idea! In mathematical terms, if we have something like A * B * C = 0, then either A = 0, B = 0, or C = 0 (or maybe even more than one of them is zero!). This is incredibly useful because it allows us to take a complex expression and break it down into simpler parts. When we look at our expression, x(x – 3)(x + 5)(x² + 4), we can see four main components being multiplied together: x, (x – 3), (x + 5), and (x² + 4). To find the values of x that make the entire expression zero, we just need to figure out what values make each of these components zero. This is where our options come into play: -5, 0, 3, and 4. We’re going to plug each of these into our expression and see what happens. By understanding the Zero Product Property, we’ve equipped ourselves with a powerful tool. Now, let's get into the nitty-gritty and see which of these values actually work. Remember, math isn't just about getting the right answer; it’s about understanding why that answer is correct. So, let’s explore this together and uncover the logic behind the zeros of this expression!

Evaluating the Options: Which Values Zero the Expression?

Okay, let's roll up our sleeves and get into the heart of the matter: evaluating each of the given options to see which ones make our expression, x(x – 3)(x + 5)(x² + 4), equal to zero. We've got four contenders: -5, 0, 3, and 4. Our strategy is straightforward – we'll substitute each value for x in the expression and watch what happens. This is where the rubber meets the road, and we’ll see the Zero Product Property in action. Remember, if any of the factors become zero, the whole expression collapses to zero. Let’s start with our first candidate: x = -5. Plugging this in, we get -5((-5) – 3)((-5) + 5)((-5)² + 4). Notice anything interesting? That (-5) + 5 part becomes zero! So, the entire expression becomes -5 * (-8) * 0 * (29), which equals zero. Bingo! So, x = -5 is definitely one of our solutions. Now, let’s move on to the next option, x = 0. This one is pretty straightforward. If we substitute x = 0, the expression becomes 0(0 – 3)(0 + 5)(0² + 4). Since the very first factor is zero, the whole thing is zero, regardless of what the other factors are. So, x = 0 is another solution. We're on a roll! Next up, we have x = 3. Substituting this in, we get 3(3 – 3)(3 + 5)(3² + 4). Again, we have a factor that becomes zero: (3 – 3). This makes the entire expression 3 * 0 * 8 * (13), which, of course, equals zero. So, x = 3 is also a solution. Three down, one to go. Finally, let’s try x = 4. Plugging this into our expression gives us 4(4 – 3)(4 + 5)(4² + 4). This simplifies to 4 * 1 * 9 * 20. Notice anything different here? None of the factors are zero! The result is 720, which is definitely not zero. So, x = 4 is not a solution. Through this step-by-step evaluation, we’ve seen exactly how each value affects the expression. The Zero Product Property has been our trusty guide, showing us that it only takes one zero factor to make the entire expression zero. We’ve identified three values that work: -5, 0, and 3. And we’ve also seen why 4 doesn’t fit the bill. This process isn't just about finding the answers; it's about understanding the underlying principles. And in this case, it’s the power of the Zero Product Property that shines through.

Deep Dive into Each Solution

Let's really get into the nitty-gritty and deep dive into each solution we found for the expression x(x – 3)(x + 5)(x² + 4) = 0. We identified three values that work: x = -5, x = 0, and x = 3. But why do these values work, and what's special about them? Let's explore each one in detail. First up, we have x = -5. When we substitute -5 into the expression, we get -5(-5 – 3)(-5 + 5)((-5)² + 4). The magic happens in the (-5 + 5) part, which equals zero. This is a perfect example of the Zero Product Property in action. The moment one factor becomes zero, the entire product becomes zero, regardless of what the other factors are. It's like having a team where one person isn't contributing – the whole team’s effort results in nothing. In this case, the (-5 + 5) factor is that non-contributor, making the whole expression equal to zero. So, x = -5 is a clear winner. Next, let’s consider x = 0. This one is almost too easy! When we plug in 0, the expression becomes 0(0 – 3)(0 + 5)(0² + 4). The very first factor is zero, so the entire expression is zero. It doesn't matter what (0 – 3), (0 + 5), or (0² + 4) are; the zero at the beginning wipes everything out. This highlights the power of zero in multiplication – it’s like a black hole that sucks everything into nothingness. So, x = 0 is another solid solution. Finally, we have x = 3. Substituting this into the expression, we get 3(3 – 3)(3 + 5)(3² + 4). Here, the (3 – 3) part is the key. It equals zero, making the entire expression 3 * 0 * 8 * 13, which is zero. Again, the Zero Product Property is our guiding light. One zero factor, and the whole product collapses. It’s like a domino effect – one zero factor knocks down the entire expression. Now, let's spare a thought for x = 4, which didn’t work. When we plugged it in, we got 4(4 – 3)(4 + 5)(4² + 4), which simplifies to 4 * 1 * 9 * 20. None of these factors are zero, so the result is a non-zero number (720, to be exact). This is a great illustration of why the Zero Product Property is so crucial – it’s not enough for most factors to be non-zero; all factors need to be non-zero for the product to be non-zero. By examining each solution in detail, we've not only confirmed our answers but also deepened our understanding of the Zero Product Property and how it governs the behavior of expressions like this. It's not just about the answers; it's about the journey of discovery and understanding the