Finding Polynomial Function With Real Coefficients Zeros 4, 4, 1+i

In the realm of mathematics, specifically within algebra, a common task involves constructing polynomial functions that satisfy certain conditions. One such condition is having a predefined set of zeros. This article delves into the process of finding a polynomial function with real coefficients when its zeros are provided. We will explore the underlying principles, address the significance of complex conjugate pairs, and demonstrate the practical steps involved in constructing such polynomials. Let's consider the zeros 4, 4, and 1+i as an example to illustrate the process. The objective is to determine a polynomial function f(x) that possesses these specific roots.

Understanding the Fundamentals of Polynomial Functions

To effectively address the task of finding a polynomial function with specified zeros, it's essential to have a firm grasp of the fundamental concepts related to polynomials. A polynomial is an expression consisting of variables and coefficients, combined using addition, subtraction, and non-negative integer exponents. The degree of a polynomial is determined by the highest power of the variable present in the expression. For instance, a quadratic equation is a polynomial of degree 2, while a cubic equation is a polynomial of degree 3. Understanding these basics is crucial for constructing polynomials with desired characteristics.

The Factor Theorem and Its Significance

The Factor Theorem plays a pivotal role in the process of constructing polynomials from known zeros. This theorem states that if r is a zero of a polynomial function f(x), then (x - r) is a factor of f(x). Conversely, if (x - r) is a factor of f(x), then r is a zero of f(x). This theorem provides a direct link between the zeros of a polynomial and its factors, enabling us to build the polynomial by multiplying these factors together. For example, if we know that 2 is a zero of a polynomial, then (x - 2) must be one of its factors. The Factor Theorem is a cornerstone in polynomial algebra, enabling us to transition smoothly between roots and factors.

Complex Conjugate Pairs: A Critical Consideration

When dealing with polynomials that have real coefficients, a crucial concept to understand is that complex zeros always occur in conjugate pairs. This means that if a complex number a + bi is a zero of the polynomial, where a and b are real numbers and i is the imaginary unit, then its complex conjugate a - bi must also be a zero. This property stems from the nature of polynomial equations with real coefficients and the way complex numbers interact in algebraic manipulations. Ignoring this principle can lead to incorrect polynomial constructions. In our example, since 1 + i is a zero, its conjugate 1 - i must also be a zero.

Multiplicity of Zeros: Accounting for Repeated Roots

The multiplicity of a zero refers to the number of times it appears as a root of the polynomial equation. If a zero r has a multiplicity of n, it means that the factor (x - r) appears n times in the factored form of the polynomial. For instance, if a zero has a multiplicity of 2, it is considered a repeated root. Recognizing the multiplicity of zeros is vital because it directly impacts the degree of the polynomial and its behavior near the corresponding x-intercept. In the given example, the zero 4 appears twice, indicating a multiplicity of 2, which we must account for when constructing the polynomial. Failing to consider the multiplicity can result in a polynomial of the wrong degree or with incorrect behavior at the repeated root.

Constructing the Polynomial Function

Now, let's apply these principles to construct the polynomial function f(x) with real coefficients that has the zeros 4, 4, and 1+i. The step-by-step process involves identifying all the zeros (including complex conjugates), applying the Factor Theorem, and multiplying the resulting factors. This approach ensures that the polynomial we construct satisfies all the given conditions.

Identifying All Zeros: Completing the Set

The first step in constructing the polynomial is to ensure we have identified all the zeros, including any complex conjugates. We are given the zeros 4, 4, and 1 + i. As discussed earlier, since 1 + i is a complex zero, its conjugate 1 - i must also be a zero. Therefore, the complete set of zeros for our polynomial function is 4, 4, 1 + i, and 1 - i. This comprehensive set forms the foundation for the subsequent steps.

Applying the Factor Theorem: Forming the Factors

Next, we apply the Factor Theorem to convert each zero into its corresponding factor. For the zero 4, the factor is (x - 4). Since 4 appears twice (multiplicity of 2), we have the factor (x - 4) squared, or (x - 4)². For the complex zero 1 + i, the factor is (x - (1 + i)) or (x - 1 - i). For its conjugate 1 - i, the factor is (x - (1 - i)) or (x - 1 + i). These factors are the building blocks of our polynomial function, derived directly from the zeros using the Factor Theorem. The correct application of this step is critical for obtaining the desired polynomial.

Multiplying the Factors: Building the Polynomial

Now, we multiply these factors together to construct the polynomial function. We have f(x) = (x - 4)²(x - 1 - i)(x - 1 + i). First, let's expand (x - 4)², which gives us x² - 8x + 16. Next, we multiply the factors corresponding to the complex conjugates: (x - 1 - i)(x - 1 + i). This multiplication results in x² - 2x + 2. Now, we multiply these two quadratic expressions: (x² - 8x + 16)(x² - 2x + 2). Expanding this product yields the polynomial f(x) = x⁴ - 10x³ + 34x² - 48x + 32. This polynomial function has real coefficients and the specified zeros, fulfilling the requirements of the problem.

Verifying the Solution and Exploring Alternative Forms

After constructing the polynomial, it's prudent to verify that it indeed has the given zeros. This can be done by substituting each zero into the polynomial and confirming that the result is zero. Additionally, we can explore alternative forms of the polynomial, such as multiplying the polynomial by a constant. This does not change the zeros but scales the polynomial vertically. Understanding these variations provides a more complete picture of the solution space.

Substituting Zeros to Verify

To verify our solution, we can substitute each zero into the polynomial f(x) = x⁴ - 10x³ + 34x² - 48x + 32 and check if the result is zero. Substituting x = 4 gives f(4) = 4⁴ - 10(4)³ + 34(4)² - 48(4) + 32 = 0, confirming that 4 is indeed a zero. Substituting x = 1 + i is more involved, but after careful calculation, we also find that f(1 + i) = 0, thus verifying the complex zero. These substitutions provide a crucial check on our work, ensuring the accuracy of the constructed polynomial.

Exploring Alternative Forms by Scaling

It's important to recognize that there are infinitely many polynomial functions that satisfy the given conditions. We can obtain alternative forms of the polynomial by multiplying it by a non-zero constant. For example, 2(x⁴ - 10x³ + 34x² - 48x + 32) is also a valid solution. Scaling the polynomial in this way does not change its zeros but alters the leading coefficient and the overall vertical scale of the graph. This flexibility is a key aspect of polynomial construction, highlighting that the solution is not unique but rather a family of polynomials sharing the same zeros.

Conclusion: Mastering Polynomial Construction

In conclusion, finding a polynomial function with real coefficients given specific zeros is a fundamental skill in algebra. The process involves understanding the Factor Theorem, recognizing the importance of complex conjugate pairs, accounting for the multiplicity of zeros, and systematically multiplying the corresponding factors. By following these steps, we can construct polynomials that meet the specified criteria. The example of finding a polynomial with zeros 4, 4, and 1 + i demonstrates the practical application of these concepts. Moreover, the ability to verify the solution and explore alternative forms enhances our understanding of polynomial functions and their properties. Mastering these techniques is invaluable for various mathematical applications and problem-solving scenarios.