Polynomial Function With Leading Coefficient 1 And Roots 2i And 3i

A crucial aspect of polynomial functions lies in their roots and leading coefficients. The roots of a polynomial function are the values of x for which the function equals zero, while the leading coefficient is the coefficient of the term with the highest degree. Understanding these elements is vital for constructing and analyzing polynomial functions. In this article, we will delve into the process of identifying a polynomial function given its roots and leading coefficient. Specifically, we aim to determine the polynomial function that has a leading coefficient of 1 and roots 2i and 3i, each with a multiplicity of 1. This exploration will involve understanding the relationship between roots and factors of a polynomial, as well as the role of complex roots in polynomial construction. By working through this problem, we'll gain a deeper appreciation for the fundamental principles governing polynomial functions.

Understanding Polynomial Functions and Their Roots

Polynomial functions, cornerstones of algebra and calculus, are expressions involving variables raised to non-negative integer powers. A deep comprehension of these functions necessitates understanding their roots, which are the values of x that render the function equal to zero. These roots, also known as zeros or solutions, play a pivotal role in defining the behavior and characteristics of the polynomial function. The relationship between roots and factors is fundamental: if a polynomial function f(x) has a root r, then (x - r) is a factor of f(x). This principle is foundational for constructing polynomial functions from their roots. Complex roots, often encountered in polynomial equations, come in conjugate pairs. If a + bi is a root of a polynomial with real coefficients, then its conjugate a - bi is also a root. This property is crucial when dealing with polynomials that have complex solutions. The multiplicity of a root indicates the number of times a particular root appears as a solution of the polynomial equation. A root with multiplicity n means the corresponding factor appears n times in the factored form of the polynomial. This concept is vital for accurately constructing polynomials with specified root characteristics. The leading coefficient of a polynomial function, the coefficient of the term with the highest degree, significantly influences the function's end behavior and overall shape. A leading coefficient of 1 simplifies the polynomial structure and often leads to more straightforward analysis. Grasping these core concepts is paramount for tackling problems involving polynomial functions, including constructing a polynomial from its given roots and leading coefficient.

Constructing Polynomial Functions from Roots

Constructing polynomial functions from their roots is a fundamental skill in algebra. The relationship between roots and factors forms the basis of this process: for every root r of a polynomial, there exists a corresponding factor (x - r). This principle enables us to build the polynomial function by multiplying these factors together. When complex roots are involved, we must consider their conjugates. Complex roots of polynomials with real coefficients always occur in conjugate pairs. Therefore, if a + bi is a root, then a - bi must also be a root. This ensures that the resulting polynomial has real coefficients, which is a common requirement in many problems. The multiplicity of a root is crucial in determining the correct form of the polynomial. A root with multiplicity n means that the corresponding factor appears n times. For instance, if a root r has multiplicity 2, the factor (x - r) appears twice, or (x - r)^2, in the polynomial. The leading coefficient plays a significant role in the final form of the polynomial. If the leading coefficient is specified as 1, the product of the factors should expand to a polynomial where the coefficient of the highest degree term is 1. If the leading coefficient is different from 1, we need to multiply the entire polynomial by a constant factor to achieve the desired leading coefficient. By systematically applying these principles, we can construct a polynomial function that precisely matches the given roots, multiplicities, and leading coefficient. This process is essential for solving various problems in algebra and calculus.

Step-by-Step Solution to the Problem

Let's solve the problem of finding the polynomial function with a leading coefficient of 1 and roots 2i and 3i, each with multiplicity 1, step by step.

1. Identify the Roots and Their Multiplicities

The given roots are 2i and 3i, both with multiplicity 1. Since the polynomial has real coefficients, the complex conjugates of these roots, which are -2i and -3i, must also be roots.

2. Form the Factors

For each root, we form a factor of the form (x - r). Thus, the factors corresponding to the roots 2i, 3i, -2i, and -3i are (x - 2i), (x - 3i), (x + 2i), and (x + 3i), respectively.

3. Construct the Polynomial

Multiply the factors together to form the polynomial function:

f(x) = (x - 2i)(x - 3i)(x + 2i)(x + 3i)

4. Simplify the Expression

Group the conjugate pairs and multiply them:

f(x) = [(x - 2i)(x + 2i)][(x - 3i)(x + 3i)]

Recall that (a - b)(a + b) = a^2 - b^2. Apply this to both pairs:

f(x) = [x^2 - (2i)^2][x2 - (3i)^2]

Since i^2 = -1:

f(x) = [x^2 - (-4)][x^2 - (-9)] f(x) = (x^2 + 4)(x^2 + 9)

Now, expand the product:

f(x) = x^4 + 9x^2 + 4x^2 + 36 f(x) = x^4 + 13x^2 + 36

5. Verify the Leading Coefficient

The leading coefficient of the resulting polynomial is 1, which matches the given condition.

Therefore, the polynomial function is:

f(x) = x^4 + 13x^2 + 36

This step-by-step solution demonstrates how to construct a polynomial function from its roots and leading coefficient, paying careful attention to complex conjugates and multiplicities.

Analyzing the Correct Option

To identify the correct option, we need to compare the derived polynomial function with the given choices. Recall that we found the polynomial function to be:

f(x) = x^4 + 13x^2 + 36

Let's analyze the given options:

A. f(x) = (x - 2i)(x - 3i) This option only considers the roots 2i and 3i and does not include their complex conjugates. It would result in a quadratic polynomial with complex coefficients, which is not what we are looking for.

B. f(x) = (x + 2i)(x + 3i) Similar to option A, this option also does not include the complex conjugates and would result in a quadratic polynomial with complex coefficients.

C. f(x) = (x - 2)(x - 3)(x - 2i)(x - 3i) This option includes real roots (2 and 3) in addition to the complex roots. This would result in a polynomial of degree 4, but it is not the correct polynomial since it has different roots than specified.

D. f(x) = (x + 2i)(x + 3i)(x - 2i)(x - 3i) This option includes the complex roots 2i and 3i along with their conjugates -2i and -3i. This is the correct set of factors. Multiplying these factors:

f(x) = (x + 2i)(x - 2i)(x + 3i)(x - 3i) f(x) = (x^2 - (2i)^2)(x2 - (3i)^2) f(x) = (x^2 + 4)(x^2 + 9) f(x) = x^4 + 13x^2 + 36

This matches the polynomial function we derived. Therefore, the correct option is D.

Conclusion

In conclusion, the polynomial function with a leading coefficient of 1 and roots 2i and 3i, each with multiplicity 1, is:

f(x) = x^4 + 13x^2 + 36

This corresponds to option D, which is f(x) = (x + 2i)(x + 3i)(x - 2i)(x - 3i). The process of solving this problem highlights the importance of understanding the relationship between roots, factors, and leading coefficients in polynomial functions. Complex roots always come in conjugate pairs, and each root contributes a factor to the polynomial. By systematically constructing the polynomial from its factors, we can arrive at the correct solution. This exercise reinforces the fundamental principles of polynomial algebra and provides a solid foundation for more advanced topics in mathematics.