Polynomial Function With Roots 1 And 1+i

Jul 29, 2025 by ADMIN 41 views

Polynomial Function of Lowest Degree with Lead Coefficient 1 and Roots 1 and 1+i

Hey everyone! Today, we're diving into the fascinating world of polynomial functions, specifically focusing on how to construct one with the lowest degree, a leading coefficient of 1, and given roots. This is a classic problem in algebra, and it's super important for understanding the relationship between roots and polynomial equations. We're going to tackle the question: What is the polynomial function of lowest degree with a leading coefficient of 1 and roots 1 and 1 + i? Let's break it down step by step so you guys can master this concept.

Understanding the Basics: Roots and Polynomials

Before we jump into solving the problem directly, let's make sure we're all on the same page with some key definitions. A root of a polynomial function, also known as a zero, is a value of x that makes the function equal to zero. In other words, if r is a root of f(x), then f(r) = 0. The degree of a polynomial is the highest power of the variable x in the polynomial. For example, in the polynomial x^3 - 2x^2 + x - 5, the degree is 3. The leading coefficient is the coefficient of the term with the highest power of x. In the same example, the leading coefficient is 1.

Now, here's a crucial concept: the Fundamental Theorem of Algebra. This theorem states that a polynomial of degree n has exactly n complex roots, counting multiplicities. This means a cubic polynomial (degree 3) will have three roots, a quadratic polynomial (degree 2) will have two roots, and so on. These roots can be real numbers, imaginary numbers, or complex numbers (which are a combination of real and imaginary parts). Another extremely important theorem for this discussion is the Complex Conjugate Root Theorem. This theorem states that if a polynomial with real coefficients has a complex root a + bi, then its complex conjugate a - bi is also a root. This is super important because complex roots always come in conjugate pairs when we're dealing with polynomials that have real coefficients, which is what we usually encounter in these types of problems. So, if we know that 1 + i is a root, we automatically know that 1 - i is also a root. This knowledge is crucial for constructing our polynomial.

Knowing these fundamental concepts will help us immensely in solving our problem. Remember, the goal is to find the polynomial function with the lowest degree that satisfies the given conditions. This means we want to include only the necessary roots and keep the polynomial as simple as possible. This principle of minimality is key in mathematics and helps us find the most elegant solution. So, with these tools in our arsenal, let's dive into the specifics of our problem and see how we can apply these concepts to find the polynomial function we're looking for. We'll start by identifying all the roots and then use them to construct the polynomial. Let's get to it!

Identifying All the Roots

The problem states that our polynomial function has roots 1 and 1 + i. Remember the Complex Conjugate Root Theorem we just discussed? Since our polynomial has real coefficients (and we're looking for one with a leading coefficient of 1, which is real), if 1 + i is a root, then its complex conjugate, 1 - i, must also be a root. So, we actually have three roots: 1, 1 + i, and 1 - i. This is a crucial observation because it tells us the lowest possible degree of our polynomial. Since we have three roots, the polynomial must be at least a cubic (degree 3) polynomial. We can't have a quadratic (degree 2) polynomial because that would only have two roots. This understanding of the relationship between the number of roots and the degree of the polynomial is fundamental to solving these kinds of problems. It guides us in the right direction and prevents us from trying to construct a polynomial that doesn't fit the given conditions. Now that we've identified all the roots, the next step is to use them to build the polynomial. We'll do this by working backwards from the roots to the factors of the polynomial, and then multiplying those factors together. This process might seem a little abstract at first, but it's a powerful technique for constructing polynomials from their roots. So, let's move on to the next step and see how we can turn these roots into a polynomial function. Remember, we're looking for the polynomial with the lowest degree and a leading coefficient of 1, so we'll keep that in mind as we build our polynomial.

Constructing the Polynomial from Its Roots

Now that we know our roots are 1, 1 + i, and 1 - i, we can start building our polynomial. Remember, if r is a root of a polynomial, then (x - r) is a factor of that polynomial. So, corresponding to the root 1, we have the factor (x - 1). For the root 1 + i, we have the factor (x - (1 + i)), which simplifies to (x - 1 - i). And for the root 1 - i, we have the factor (x - (1 - i)), which simplifies to (x - 1 + i). To get our polynomial, we multiply these factors together: f(x) = (x - 1)(x - 1 - i)(x - 1 + i). The next step is to simplify this expression. Notice that we have a pair of complex conjugate factors: (x - 1 - i) and (x - 1 + i). Multiplying complex conjugates is a neat trick because the imaginary parts cancel out, leaving us with a real quadratic expression. Let's multiply these two factors first:

(x - 1 - i)(x - 1 + i) = ((x - 1) - i)((x - 1) + i)

This looks like the difference of squares: (a - b)(a + b) = a^2 - b^2, where a = (x - 1) and b = i. So, we have:

((x - 1)^2 - i^2) = (x^2 - 2x + 1 - (-1)) = x^2 - 2x + 2

Now we have a much simpler expression. Our polynomial is now:

f(x) = (x - 1)(x^2 - 2x + 2)

Finally, we multiply this out to get the polynomial in standard form:

f(x) = x(x^2 - 2x + 2) - 1(x^2 - 2x + 2) = x^3 - 2x^2 + 2x - x^2 + 2x - 2 = x^3 - 3x^2 + 4x - 2

So, the polynomial function of lowest degree with a leading coefficient of 1 and roots 1 and 1 + i is f(x) = x^3 - 3x^2 + 4x - 2. We've successfully constructed the polynomial from its roots, making use of the Complex Conjugate Root Theorem and simplifying the expression step by step. This process highlights the powerful connection between roots and factors in polynomial functions. Now, let's take a look at the answer choices and see which one matches our result.

Matching the Answer Choices

We found that the polynomial function is f(x) = x^3 - 3x^2 + 4x - 2. Now, let's compare this to the answer choices provided:

A. f(x) = x^2 - 2x + 2 B. f(x) = x^3 - x^2 + 4x - 2 C. f(x) = x^3 - 3x^2 + 4x - 2 D. f(x) = x^2 - x + 2

By direct comparison, we can see that option C, f(x) = x^3 - 3x^2 + 4x - 2, matches our result exactly. Therefore, the correct answer is C. It's always a good idea to double-check your work, especially in math problems. We can quickly verify that our polynomial has the correct roots by plugging them in. If we plug in x = 1, we get:

f(1) = 1^3 - 3(1)^2 + 4(1) - 2 = 1 - 3 + 4 - 2 = 0

So, 1 is indeed a root. Plugging in x = 1 + i would be a bit more involved, but we know we constructed the polynomial specifically to have 1 + i and 1 - i as roots, so we can be confident in our answer. This process of verifying the solution reinforces our understanding and reduces the chance of errors. So, we've successfully found the polynomial function, matched it to the answer choices, and even verified our result. Let's wrap up with a quick recap of the key steps we took.

Recap and Key Takeaways

Alright guys, let's quickly recap what we've done in this problem. We started with the question: What is the polynomial function of lowest degree with a leading coefficient of 1 and roots 1 and 1 + i? To solve this, we first understood the importance of the Complex Conjugate Root Theorem, which told us that if 1 + i is a root, then 1 - i must also be a root. This gave us a total of three roots: 1, 1 + i, and 1 - i. We then used these roots to construct the factors of the polynomial: (x - 1), (x - 1 - i), and (x - 1 + i). Multiplying these factors together, we simplified the expression by first multiplying the complex conjugate factors to eliminate the imaginary parts. This led us to the polynomial f(x) = x^3 - 3x^2 + 4x - 2. Finally, we compared our result to the answer choices and found that option C matched perfectly. We even took a moment to verify that 1 is indeed a root of our polynomial. The key takeaways from this problem are:

Understanding the relationship between roots and factors of a polynomial.
Knowing and applying the Complex Conjugate Root Theorem.
Simplifying expressions involving complex numbers.
Constructing a polynomial from its roots.

These are fundamental skills in algebra, and mastering them will help you tackle a wide range of polynomial problems. Remember, the lowest degree requirement is crucial because it guides us to include only the necessary roots. The leading coefficient of 1 simplifies the process, but the same principles apply even with different leading coefficients. So, keep practicing, and you'll become a polynomial pro in no time! Understanding how to construct polynomials from their roots is a powerful tool in your mathematical arsenal. It allows you to move seamlessly between roots and equations, providing a deeper insight into the nature of polynomial functions. Keep these concepts in mind, and you'll be well-prepared to tackle similar problems in the future.

Correct Answer: C