Find The Complex Zeros Of The Following Polynomial Function
Ever stare at a polynomial and think, "Wow, that looks like a complicated party I'm not invited to"? You know, those stringy things with exponents like x3 + 2x2 - 5x + 1? Sometimes, mathematicians, in their infinite wisdom (or perhaps just for a good brain-bending challenge), want to know where these functions hit rock bottom, or perhaps soar to unexpected heights. We call these points the "zeros" of the function. Think of it like finding the exact spot where a rollercoaster track touches the ground. Easy peasy, right?
Well, sometimes those zeros are as plain as day, hiding in plain sight like a lost sock. But then there are the sneaky ones. The ones that don't want to play by the rules of regular numbers. These, my friends, are the "complex zeros". Imagine trying to find a treasure chest, and the map says the treasure is buried "somewhere over the rainbow." That's kind of what finding complex zeros can feel like initially. It's like venturing into a whole new dimension of numbers!
Now, let's say we have a particularly rambunctious polynomial staring us down. This one is a real show-off, and it's got some serious roots we need to unearth. We're talking about this magnificent beast: f(x) = x4 + 5x2 + 4. Doesn't it just look like a mathematical masterpiece, just begging to be understood? It's like a grand old mansion with hidden passages and secret rooms. Our mission, should we choose to accept it (and we totally should!), is to unlock all those hidden chambers and find every single one of its zeros, even the ones that are a little bit... exotic.
So, picture this polynomial. It's a bit of a mystery, isn't it? It's not just your average, everyday, "plug in a number and get a number out" kind of deal. This one has a flair for the dramatic. And when we're looking for its complex zeros, we're essentially looking for the keys to unlock its deepest secrets. It's like being a detective on a thrilling case, and the clues are all mathematical!
Now, the first thing you might notice about our particular polynomial, f(x) = x4 + 5x2 + 4, is that it's a bit shy about showing off its odd powers. It's like a performer who only likes to wear their fanciest outfits for the even-numbered acts. Notice there's no x3 or x1 term. This little tidbit is a HUGE clue, a shimmering hint from the mathematical universe itself! It’s like finding a secret handshake in a crowd of strangers.
Because of this fascinating pattern, we can treat this polynomial like a much simpler one. Imagine you're trying to solve a really big puzzle, but you realize a whole section of it looks exactly like a smaller, easier puzzle you've already solved! That's exactly what's happening here. We can make a clever substitution. Let's pretend that x2 is actually just a regular, friendly letter, say, 'y'. So, instead of x4 + 5x2 + 4, we now have a much more manageable-looking expression: y2 + 5y + 4. Ta-da! It's like a magician pulling a rabbit out of a hat, but the rabbit is a much simpler equation.
Now, this new expression, y2 + 5y + 4, is a classic quadratic. It's like that comfortable old armchair in your living room. You know exactly how to work with it. We can factor it! Think of it like finding two numbers that multiply to 4 and add up to 5. Any guesses? Of course! It’s 4 and 1. So, we can rewrite our 'y' equation as (y + 4)(y + 1) = 0. See? It's all coming together beautifully!
But wait! We’re not done yet. Remember, our original quest was for the zeros of f(x), not for the zeros of 'y'. So, we need to substitute back. We know that y = x2. So, our factored equation now becomes (x2 + 4)(x2 + 1) = 0. We've peeled back another layer of the onion, and we're getting closer to the core!
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Now, for each of these factors, we set them equal to zero. This is where things get really interesting, where the magic truly happens! First, let's tackle x2 + 4 = 0. To solve for x here, we'll subtract 4 from both sides, giving us x2 = -4. Uh oh! What number, when squared, gives you a negative result? In the world of regular numbers, this is a no-go. It's like trying to find a square peg for a round hole. But remember those complex zeros we were talking about? This is their moment to shine!
This is where the magical number 'i' enters the stage! The imaginary unit, 'i', is defined as the square root of -1. It's the key that unlocks these previously inaccessible zeros. It's the secret ingredient that makes the mathematical cake rise!
Solved Find the complex zeros of the following polynomial | Chegg.com
So, when we take the square root of both sides of x2 = -4, we get x = ±√(-4). And thanks to our friend 'i', this becomes x = ±√4 * √(-1), which simplifies to x = ±2i. Boom! Two of our complex zeros right there, dazzling and daring!
Now, for the grand finale! We still have our other factor to conquer: x2 + 1 = 0. Let's do the same dance. Subtract 1 from both sides to get x2 = -1. And what do we know about x2 = -1? That's right! That's the very definition of our friend 'i'! So, taking the square root of both sides, we get x = ±√(-1), which, in our new, expanded number system, is simply x = ±i. Another two zeros, elegantly revealed!
So, there you have it! The complex zeros of our magnificent polynomial f(x) = x4 + 5x2 + 4 are 2i, -2i, i, and -i. Aren't they just a joy? They might not be on the familiar number line, but they are absolutely crucial to understanding the complete picture of this mathematical function. It's like discovering that the characters in your favorite story have a whole secret lives you never knew about! Finding these complex zeros isn't just about solving an equation; it's about expanding our understanding of what numbers can do, and that, my friends, is a truly exciting adventure.