Finding Polynomials: Zeros, Coefficients, And Expanded Forms

Hey guys! Let's dive into some cool math problems. We're going to explore how to build polynomials based on their zeros and leading coefficients. It's like a puzzle, and we get to put the pieces together. I will address Question 6 and Question 7, so let's get started. Get ready to flex those math muscles! We will learn how to deal with complex numbers and real coefficients, which is super useful in algebra and beyond. This is going to be fun, I promise!

Question 6: Constructing a Degree 3 Polynomial

Finding a degree 3 polynomial with real coefficients having zeros 1 and 2i, and a lead coefficient of 1. This is a classic problem, so let's break it down step by step to see how we tackle it. When we are told we need a polynomial with real coefficients, and one of the zeros is a complex number, we immediately know something important: complex roots come in conjugate pairs. If 2i is a zero, then its conjugate, -2i, must also be a zero. This is a fundamental concept in polynomial theory, so make sure to keep this in mind.

Since we're looking for a degree 3 polynomial, we know it will have three zeros (counting multiplicities). We already have two zeros: 1 and 2i. Thanks to the conjugate root theorem, we know the third zero is -2i. So, our zeros are 1, 2i, and -2i. The lead coefficient is given as 1. This is super handy because it simplifies our calculations, meaning that we won't need to do any scaling later on. To build the polynomial, we use the zeros to write factors. A zero of 1 gives us a factor of (x - 1). A zero of 2i gives us a factor of (x - 2i). A zero of -2i gives us a factor of (x + 2i). Therefore, our polynomial can be written as:

P(x) = 1 * (x - 1) * (x - 2i) * (x + 2i)

Now we'll work on expanding this to get it into the standard form. First, let's multiply the complex conjugate factors: (x - 2i) * (x + 2i). When we multiply these, we get x² + 2ix - 2ix - 4i². The middle terms cancel out, and remember that i² = -1. So, this simplifies to x² - 4(-1), which is x² + 4. Now we can rewrite our polynomial as:

P(x) = (x - 1) * (x² + 4)

Next, multiply out the remaining factors: (x - 1) * (x² + 4). Distribute the x: x * x² = x³ and x * 4 = 4x. Then, distribute the -1: -1 * x² = -x² and -1 * 4 = -4. Combining these, we get x³ - x² + 4x - 4. So the expanded form of our polynomial is:

P(x) = x³ - x² + 4x - 4

And that's it! We have successfully found a degree 3 polynomial with the specified zeros and lead coefficient, and we've written it in the expanded form. This process is key for many applications. This is how it's done. Understanding this is crucial, and you can see how the conjugate pairs pop up anytime you're dealing with a polynomial with real coefficients. It all comes together nicely.

Question 7: Another Degree 3 Polynomial Challenge

Alright, let's try a similar problem. Find a degree 3 polynomial with real coefficients. The goal is to build our polynomial from scratch, using the given information to construct it step-by-step. The process is similar to Question 6, but we'll adapt to any new information. We are given different information, that is the difference with question 6. This is where we show our problem-solving skills and demonstrate how to handle more complex scenarios.

We need to find a degree 3 polynomial with real coefficients, but this time, the problem doesn't directly give us the zeros. Instead, we have some clues that will lead us to the correct zeros. The information we have includes other constraints or relationships that can help us. The more we understand about polynomials, the more tricks we have to solve these problems. We might have some roots, like we did in the last problem. We are going to have to use our knowledge of roots to determine the rest of the polynomials. Remember the key concept of the conjugate pair theorem when working with complex roots. If a complex number is a root, so is its conjugate.

The strategy is to find the zeros, knowing this will help in this case. Once we have the zeros, we will construct the polynomial using the same approach as before. This also means we'll work with the factors and then expand the factors. We may use the conjugate pair theorem if we encounter any complex roots.

Let's assume, for the sake of the exercise, that this degree 3 polynomial has a root. We know that complex roots come in conjugate pairs, so if we can find one complex root, we know another. We'll use our knowledge of factoring and polynomial division, which is critical here.

Let's work through an example of how we find these polynomials: Suppose we are told that 2 + i is a zero, and the lead coefficient is 1. If 2 + i is a zero, then the conjugate, 2 - i, is also a zero. These are a pair of zeros. With these complex conjugate zeros, and because we know this is a degree 3 polynomial, we need to find one more real root to get our polynomial. Let's assume the other real root is 3. Now we know all three roots: 2 + i, 2 - i, and 3. We can use these to build our polynomial.

First, we create our factors. From the root of 2 + i, we get the factor (x - (2 + i)). From the root of 2 - i, we get the factor (x - (2 - i)). From the root of 3, we get (x - 3). Because the leading coefficient is 1, our polynomial will be:

P(x) = (x - (2 + i)) * (x - (2 - i)) * (x - 3)

Now, we multiply the complex conjugate factors: (x - (2 + i)) * (x - (2 - i)). This can be rewritten as ((x - 2) - i) * ((x - 2) + i). Multiplying these gives us (x - 2)² - i². Simplifying this gives us (x² - 4x + 4) - (-1) which equals x² - 4x + 5. Now, multiply this result by (x - 3)

(x² - 4x + 5) * (x - 3)

So: x³ - 3x² - 4x² + 12x + 5x - 15. Simplify to get x³ - 7x² + 17x - 15. Therefore, the degree 3 polynomial with a lead coefficient of 1, and with zeros 2 + i, 2 - i, and 3, is:

P(x) = x³ - 7x² + 17x - 15

Conclusion: Polynomial Power!

There you have it! We've successfully navigated through the process of finding and expanding polynomials, working with complex roots, and using the conjugate pair theorem. We've tackled the problems step-by-step, making sure we understood the logic and the reasoning behind each step. I hope this was helpful! We've also learned about the importance of conjugate pairs and how they arise in problems with real coefficients. Keep practicing, and you'll become a polynomial master in no time! Remember, every problem is a chance to learn, so embrace the challenge and enjoy the journey!