Finding The Right Polynomial: A Step-by-Step Guide

Hey math enthusiasts! Let's dive into a cool polynomial problem. We're on a quest to find the perfect polynomial function, one with a leading coefficient of 1 and some specific roots: 2i and 3i, each with a multiplicity of 1. Don't worry, it's not as scary as it sounds! We'll break it down step by step and make sure we understand the logic behind each move. So, grab your pencils and let's get started. This is not just about finding an answer; it's about understanding why the answer is the way it is. By the end, you'll not only solve the problem, but also boost your understanding of polynomial functions.

Understanding the Basics: Polynomials, Roots, and Coefficients

Alright, before we jump into the options, let's quickly review the essentials. A polynomial function is just a function that involves only non-negative integer powers of a variable, like x, along with some coefficients. Think of it as a sum of terms, where each term is a constant multiplied by a power of x. The leading coefficient is the number that sits in front of the term with the highest power of x. In our case, we want this to be 1, which means our polynomial starts with something like x to the power of something or just x itself.

Now, let's talk about roots. The roots of a polynomial are the x-values where the function equals zero. They're where the graph of the polynomial crosses the x-axis. In this problem, we're given the roots 2i and 3i. Remember, i is the imaginary unit, where i² = -1. This means our roots are complex numbers, and they will influence the shape and behavior of our polynomial.

Finally, multiplicity tells us how many times a particular root appears. A multiplicity of 1 means each root shows up only once. If a root has a multiplicity of 2, it would show up twice, and so on. For our question, the roots 2i and 3i both have a multiplicity of 1, meaning each root appears only once in the factored form of the polynomial.

To make sure we're all on the same page, let's recap: We need a polynomial with a leading coefficient of 1, and roots 2i and 3i, each appearing only once. With these fundamentals, we're well-equipped to tackle the multiple-choice options.

Decoding the Multiple-Choice Options: A Close Look

Now, let's put on our detective hats and examine each option to see which one fits the bill. We'll methodically go through each choice, breaking down why it does or doesn't meet our criteria. Remember, we need a leading coefficient of 1, and the roots 2i and 3i with a multiplicity of 1.

Option A: $f(x) = (x + 2i)(x + 3i)(x - 2i)(x - 3i)$

In this option, we see a product of four factors. Let's analyze the roots. If we set each factor equal to zero, we find roots of -2i, -3i, 2i, and 3i. Oops! We see that the polynomial includes both 2i and 3i as roots, but also their negatives. Also, since there are four factors, this implies that the degree of the polynomial is four (because we'd multiply four x's together), but we only need two roots with multiplicity 1. This option would also produce a leading coefficient of 1 because each x term is multiplied by 1. However, since it contains the complex conjugates -2i and -3i, it's not quite what we are looking for. So, this option doesn't quite match our requirements.

Option B: $f(x) = (x - 2i)(x - 3i)$

This option appears promising at first glance. It has two factors, which would mean our polynomial is quadratic (degree 2). Setting each factor to zero, we find roots of 2i and 3i, which is great. And, if we expand this out, the leading coefficient will indeed be 1, as the x terms will multiply to form x². Also, the multiplicity is 1 for each root, since each root shows up once. Because all requirements are met, it appears as the best choice. This option seems to fit the bill perfectly, as it includes the roots we need and it has the right leading coefficient. This is indeed a strong contender!

Option C: $f(x) = (x - 2)(x - 3)(x - 2i)(x - 3i)$

Looking at option C, we see a more complex structure, with four factors. This means our polynomial would have a degree of 4, the xs would multiply together to produce x to the fourth power. Setting each factor equal to zero, we see that the roots would be 2, 3, 2i, and 3i. While it includes our target roots 2i and 3i, it also includes the real roots 2 and 3. This isn't what we're looking for, as we only need the imaginary roots and their respective multiplicity of 1. And the leading coefficient would be 1.

Option D: $f(x) = (x + 2i)(x + 3i)$

This option is similar to Option B, but we'll still carefully analyze it. We see two factors, so our degree is 2. Setting each factor to zero, we find roots of -2i and -3i. This means our roots are the complex conjugates of what we are looking for. Therefore, this option doesn't match our specific requirements for roots. If we were to expand this, our leading coefficient would be 1.

The Verdict: Selecting the Correct Answer

After a thorough analysis of each option, the clear winner is Option B. Here's why:

• Correct Roots: It includes the roots 2i and 3i. We are looking for these roots. Both roots have a multiplicity of 1.
• Correct Leading Coefficient: The leading coefficient is 1.

Option B provides precisely what we're looking for, without any unnecessary complications. That's the one we should choose. Options A, C, and D are incorrect because they either include extra roots, or have roots that are the complex conjugates of our target roots, or both.

Expanding Your Knowledge: Key Takeaways and Further Exploration

Well done, guys! You've successfully navigated a polynomial problem, showcasing a strong grasp of the fundamentals. Let's recap some key takeaways:

• Understanding Roots: Roots are the x-values where a polynomial equals zero. Complex roots like 2i and 3i come in pairs if the polynomial has real coefficients, often with a corresponding conjugate root. In our case, the negative roots were not included in the correct answer.
• Multiplicity Matters: Multiplicity tells us how many times a root appears. Each root in this problem had a multiplicity of 1.
• Leading Coefficient: The leading coefficient determines the end behavior and overall shape of the polynomial. A leading coefficient of 1, as required in this case, simplifies the equation to some degree.
• Building Polynomials: You can construct a polynomial from its roots by using the factored form. For each root r, you include a factor of (x - r). Multiply all these factors together, and you get your polynomial.

To solidify your understanding, try these activities:

• Practice with Different Roots: Work through similar problems with different roots and leading coefficients. Experiment with real and complex roots, and change the multiplicity of the roots.
• Graphing Polynomials: Use graphing calculators or software to visualize the polynomials. See how the roots and leading coefficients affect the graph's shape. This is an awesome way to help visually understand the solutions.
• Explore Complex Conjugates: Investigate how complex roots always come in conjugate pairs when working with polynomials that have real coefficients.

Keep practicing, and keep exploring. Math is all about discovery, so embrace the challenge and enjoy the journey!