Find Zeros Of Polynomial: P(x) = X³ + 6x² + 21x + 26
Hey guys! Let's dive into finding all the zeros of the polynomial P(x) = x³ + 6x² + 21x + 26, given that one of the zeros is -2 - 3i. Polynomial problems can seem tricky, but with a step-by-step approach, we can totally nail this. So grab your pencils, and let's get started!
Understanding Complex Conjugate Root Theorem
Before we jump into the calculations, it's super important to understand the Complex Conjugate Root Theorem. This theorem states that if a polynomial with real coefficients has a complex number as a root, then its complex conjugate is also a root. In simpler terms, if a + bi is a root, then a - bi is also a root. Lucky for us, our polynomial P(x) has real coefficients, which means that if -2 - 3i is a root, then its complex conjugate, -2 + 3i, must also be a root. Knowing this theorem saves us a lot of time and effort because we automatically get another zero for free!
Now, you might be wondering, why is this theorem so important? Well, it helps us narrow down the possibilities when searching for roots. Imagine if we didn't know about this theorem; we might have to use more complicated methods to find the other roots. This theorem acts as a shortcut, making our lives much easier. When dealing with polynomials, always keep this theorem in mind, especially when you spot a complex root. It's like finding a golden ticket that leads you straight to another root. Remember, this theorem only applies when the polynomial has real coefficients, so make sure to check that condition before applying the theorem. In our case, P(x) = x³ + 6x² + 21x + 26 happily meets this criterion, so we're good to go!
Finding the Quadratic Factor
Okay, so now we know that -2 - 3i and -2 + 3i are both zeros of P(x). What's next? Well, we can use these zeros to find a quadratic factor of P(x). Remember that if r is a root of a polynomial, then (x - r) is a factor. So, (x - (-2 - 3i)) and (x - (-2 + 3i)) are factors of P(x). To find the quadratic factor, we simply multiply these two factors together:
[x - (-2 - 3i)] * [x - (-2 + 3i)] = (x + 2 + 3i) * (x + 2 - 3i)
This looks a bit messy, but don't worry; we can simplify it. Notice that we have the form (a + b) * (a - b), which is equal to a² - b². In our case, a = (x + 2) and b = 3i. So, we have:
(x + 2)² - (3i)² = (x² + 4x + 4) - (9i²)
Since i² = -1, we get:
x² + 4x + 4 - 9(-1) = x² + 4x + 4 + 9 = x² + 4x + 13
Alright! So, the quadratic factor of P(x) is x² + 4x + 13. This quadratic expression divides P(x) perfectly and will help us find the remaining zeros. Think of it like breaking down a complex problem into smaller, more manageable parts. This quadratic factor encapsulates the information from the two complex roots, making it easier to find the remaining root. Make sure to double-check your calculations when finding this quadratic factor, as any error here will propagate through the rest of the solution. With this factor in hand, we're one step closer to unraveling all the secrets hidden within P(x).
Using Polynomial Division
Now that we have the quadratic factor x² + 4x + 13, we can use polynomial division to find the remaining factor of P(x). We're essentially dividing P(x) = x³ + 6x² + 21x + 26 by x² + 4x + 13. Let's perform the polynomial division:
x + 2
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x² + 4x + 13 | x³ + 6x² + 21x + 26
- (x³ + 4x² + 13x)
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2x² + 8x + 26
- (2x² + 8x + 26)
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0
The quotient we get from the division is x + 2. This means that P(x) = (x² + 4x + 13)(x + 2). Polynomial division is a fundamental tool in algebra, allowing us to break down higher-degree polynomials into simpler factors. In this case, it neatly reveals the remaining linear factor. It’s super crucial to perform the division carefully to avoid any mistakes, as the accuracy of the quotient directly impacts our ability to find the last zero. So, take your time, double-check each step, and make sure you're subtracting the terms correctly. The beauty of polynomial division is how it systematically peels away the layers of the polynomial, revealing its underlying structure. This skill is essential for anyone tackling polynomial problems, and mastering it will definitely boost your confidence in algebra.
Finding the Last Zero
From the polynomial division, we found that P(x) = (x² + 4x + 13)(x + 2). We already know the zeros from the quadratic factor (which are -2 - 3i and -2 + 3i). Now, we need to find the zero from the linear factor (x + 2). To do this, we set (x + 2) = 0 and solve for x:
x + 2 = 0
x = -2
So, the last zero of P(x) is -2. This was a pretty straightforward step, thanks to the polynomial division. It's amazing how neatly everything falls into place once we've correctly identified the factors. Finding the zeros of a polynomial is like piecing together a puzzle, and each step we take brings us closer to completing the picture. Don't underestimate the power of simple algebraic manipulations, as they often lead to elegant solutions. In this case, setting the linear factor to zero gave us the final piece of the puzzle, revealing the last zero of P(x). With this zero in hand, we can confidently say that we've found all the roots of the polynomial.
Final Answer
Alright, guys! We've successfully found all the zeros of the polynomial P(x) = x³ + 6x² + 21x + 26. The zeros are:
- -2 - 3i
- -2 + 3i
- -2
And that's it! We used the Complex Conjugate Root Theorem, found the quadratic factor, performed polynomial division, and solved for the remaining zero. Not too shabby, right? Remember, practice makes perfect, so keep working on these types of problems, and you'll become a pro in no time. Keep up the great work, and I'll see you in the next math adventure!
