Constructing A Polynomial Function With Real Coefficients Given Zeros
Finding a polynomial function with real coefficients given specific zeros involves understanding the fundamental theorem of algebra and the properties of complex conjugates. This article delves into the process of constructing such a polynomial, providing a step-by-step guide and detailed explanations to ensure clarity and comprehension. Specifically, we will address the case where the zeros are √6, 6, and 1 + i.
Understanding the Zeros and Their Implications
When tasked with finding a polynomial function, the zeros provided are crucial pieces of information. The zeros of a polynomial are the values of x for which the polynomial evaluates to zero. In other words, if r is a zero of a polynomial f(x), then f(r) = 0. Given the zeros √6, 6, and 1 + i, we can infer several key aspects about the polynomial we aim to construct.
First, the presence of √6 as a zero implies that the polynomial will have a factor of (x - √6). Similarly, the zero 6 indicates a factor of (x - 6). These are straightforward as they are real numbers. However, the complex zero 1 + i introduces an additional layer of complexity. Polynomials with real coefficients have a property that complex roots occur in conjugate pairs. This means that if 1 + i is a zero, its complex conjugate, 1 - i, must also be a zero. This is a critical point because failing to include the conjugate will result in a polynomial with complex coefficients, which contradicts the requirement for real coefficients. Therefore, we must also include a factor corresponding to the zero 1 - i, which will be (x - (1 - i)) or (x - 1 + i).
Understanding this conjugate pair property is fundamental in constructing polynomials with real coefficients. The complex conjugate arises from the quadratic formula where the square root of a negative number produces an imaginary term. If a polynomial has real coefficients, these imaginary terms must cancel out, necessitating the presence of conjugate pairs. By including both 1 + i and 1 - i as zeros, we ensure that when we expand the polynomial, the imaginary components will indeed eliminate each other, resulting in a polynomial with real coefficients. This initial comprehension of the zeros and their conjugates sets the stage for the next step, which involves forming the factors of the polynomial.
Constructing the Factors of the Polynomial
With a clear understanding of the zeros, the next step involves constructing the factors of the polynomial. Each zero corresponds to a factor in the form of (x - zero). We have the zeros √6, 6, 1 + i, and 1 - i. Therefore, we can write down the corresponding factors:
- For the zero √6, the factor is (x - √6).
- For the zero 6, the factor is (x - 6).
- For the zero 1 + i, the factor is (x - (1 + i)) which simplifies to (x - 1 - i).
- For the zero 1 - i, the factor is (x - (1 - i)) which simplifies to (x - 1 + i).
These factors are the building blocks of our polynomial. The polynomial function f(x) can be expressed as a product of these factors. That is, f(x) = a(x - √6)(x - 6)(x - 1 - i)(x - 1 + i), where a is a constant coefficient. The constant a allows for infinitely many correct answers, as any non-zero real number can be chosen for a and the zeros of the polynomial will remain unchanged. For simplicity, we often choose a to be 1, but it is important to recognize that any real number would yield a valid polynomial.
The factors involving the complex zeros, (x - 1 - i) and (x - 1 + i), are particularly important because they will produce real coefficients only when multiplied together. This is due to the conjugate nature of the complex roots. When these factors are multiplied, the imaginary parts will cancel out, resulting in a quadratic expression with real coefficients. This is a crucial step in ensuring that the final polynomial has the desired properties. The next phase in constructing the polynomial involves multiplying these factors together, starting with the complex conjugate pair to simplify the process and maintain real coefficients.
Multiplying the Factors and Simplifying
Once the factors of the polynomial are identified, the next crucial step is to multiply these factors together to obtain the polynomial function in its expanded form. This process involves careful algebraic manipulation, particularly when dealing with complex numbers. We start by multiplying the factors corresponding to the complex conjugate pair: (x - 1 - i) and (x - 1 + i). This strategy simplifies the expansion and ensures that the resulting quadratic expression has real coefficients.
Let's multiply (x - 1 - i) and (x - 1 + i):
(x - 1 - i)(x - 1 + i) = (x - 1)^2 - (i)^2
Expanding this gives:
x^2 - 2x + 1 - (-1) = x^2 - 2x + 1 + 1 = x^2 - 2x + 2
As expected, the result is a quadratic expression with real coefficients. This confirms the importance of including complex conjugate pairs when constructing polynomials with real coefficients. Now that we have simplified the complex factors, we multiply the remaining real factors together with this quadratic expression. The remaining factors are (x - √6) and (x - 6). So, we need to multiply (x - √6)(x - 6)(x^2 - 2x + 2).
First, let's multiply (x - √6) and (x - 6):
(x - √6)(x - 6) = x^2 - 6x - √6x + 6√6
Now, we multiply this result by the quadratic expression (x^2 - 2x + 2):
(x^2 - 6x - √6x + 6√6)(x^2 - 2x + 2) = x^4 - 2x^3 + 2x^2 - 6x^3 + 12x^2 - 12x - √6x^3 + 2√6x^2 - 2√6x + 6√6x^2 - 12√6x + 12√6
Combining like terms, we get:
x^4 + (-2 - 6 - √6)x^3 + (2 + 12 + 2√6 + 6√6)x^2 + (-12 - 2√6 - 12√6)x + 12√6
This simplifies to:
x^4 - (8 + √6)x^3 + (14 + 8√6)x^2 - (12 + 14√6)x + 12√6
This resulting polynomial function has real coefficients and the specified zeros. The process of multiplying and simplifying the factors demonstrates the algebraic rigor required to construct polynomials with specific characteristics. The final step involves writing the complete polynomial function.
Writing the Complete Polynomial Function
After multiplying and simplifying the factors, the final step is to express the complete polynomial function. This involves compiling the results of the previous calculations into a standard polynomial form. From our previous steps, we have the polynomial:
f(x) = x^4 - (8 + √6)x^3 + (14 + 8√6)x^2 - (12 + 14√6)x + 12√6
This polynomial function f(x) has real coefficients and the zeros √6, 6, 1 + i, and 1 - i. It is crucial to verify that the polynomial satisfies the given conditions. We can confirm that the coefficients are real numbers, and that the complex roots appear in conjugate pairs, which is a necessary condition for a polynomial with real coefficients.
It is worth noting that this is just one possible solution. As mentioned earlier, we could multiply the entire polynomial by any non-zero real number and still satisfy the conditions. For example, we could multiply the polynomial by 2, -1, or any other real number without changing the zeros of the function. This flexibility highlights the fact that there are infinitely many polynomial functions that could have the given zeros.
In summary, finding a polynomial function with real coefficients given specific zeros involves several key steps: understanding the implications of the zeros, including the complex conjugate pairs, constructing the factors of the polynomial, multiplying and simplifying these factors, and finally, writing the complete polynomial function. This process demonstrates the interplay between algebra and the properties of complex numbers, providing a comprehensive approach to constructing polynomials with desired characteristics. This methodology can be applied to a variety of similar problems, making it a valuable skill in polynomial algebra.
Find a Polynomial Function with Real Coefficients that has the Given Zeros
Finding a polynomial function with real coefficients given specific zeros is a fundamental problem in algebra. This article will guide you through the process, using the zeros √6, 6, and 1 + i as an example. We'll explore the concepts of complex conjugates and how they play a crucial role in constructing such polynomials. This comprehensive guide aims to provide a clear and detailed explanation, making the process accessible and understandable.
Understanding the Zeros and Complex Conjugates
When given a set of zeros, the first step is to understand what they imply about the polynomial. A zero of a polynomial, f(x), is a value of x for which f(x) = 0. In our case, the given zeros are √6, 6, and 1 + i. Since we need real coefficients, a crucial concept to remember is that complex zeros always come in conjugate pairs. This means that if 1 + i is a zero, its conjugate, 1 - i, must also be a zero. This property is essential because it ensures that when we construct the polynomial, the imaginary parts will cancel out, leaving us with real coefficients.
Why do complex conjugates matter? Complex numbers arise from the square root of negative numbers, often in the quadratic formula. If a polynomial has real coefficients, these imaginary parts must eventually eliminate each other. The only way to achieve this is if complex roots come in pairs that are conjugates of each other. The complex conjugate of a number a + bi is a - bi. When you add or multiply a complex number with its conjugate, the imaginary parts cancel out, resulting in a real number. This property is fundamental in ensuring that the resulting polynomial has real coefficients, meeting the initial requirement of the problem. Recognizing this from the outset is crucial because it dictates the factors we need to include when constructing the polynomial.
Having a firm grasp on the zeros and the significance of complex conjugates allows us to proceed with confidence. The zeros √6 and 6 are straightforward real roots, each contributing a corresponding linear factor. The complex root 1 + i, however, necessitates the inclusion of its conjugate, 1 - i, to maintain the real coefficients. Neglecting to include the conjugate would lead to a polynomial with complex coefficients, which would not satisfy the problem's conditions. Therefore, our set of zeros effectively expands to √6, 6, 1 + i, and 1 - i. This complete set of zeros provides the foundation for the next step: constructing the factors of the polynomial. Each zero will correspond to a factor, and the product of these factors will yield the polynomial we seek. The understanding of these foundational principles ensures that the polynomial we construct will adhere to the conditions specified in the problem statement.
Constructing the Factors
Once the zeros are identified, including the complex conjugates, the next step is to construct the factors of the polynomial. Each zero, r, corresponds to a factor of the form (x - r). Therefore, from the zeros √6, 6, 1 + i, and 1 - i, we can create the following factors:
- For the zero √6, the factor is (x - √6).
- For the zero 6, the factor is (x - 6).
- For the zero 1 + i, the factor is (x - (1 + i)), which simplifies to (x - 1 - i).
- For the zero 1 - i, the factor is (x - (1 - i)), which simplifies to (x - 1 + i).
These factors are the building blocks of the polynomial function. The polynomial, f(x), can be expressed as a product of these factors multiplied by a constant a, which can be any non-zero real number. This constant allows for an infinite number of correct answers, as multiplying the entire polynomial by a constant doesn't change its zeros. We can represent the polynomial as: f(x) = a(x - √6)(x - 6)(x - 1 - i)(x - 1 + i). The constant a provides a degree of freedom, allowing us to scale the polynomial vertically without altering the roots. For simplicity, we often choose a to be 1, but it’s crucial to recognize that any non-zero real value would yield a valid polynomial meeting the specified criteria.
The factors corresponding to the complex zeros, (x - 1 - i) and (x - 1 + i), are particularly significant because they will combine to form a quadratic expression with real coefficients. This is a direct consequence of the complex conjugate property. Multiplying these factors together will eliminate the imaginary components, resulting in a real-coefficient quadratic. This strategic grouping of conjugate factors is a common technique in polynomial construction, ensuring that the final expression adheres to the problem's requirements. The next logical step involves multiplying these factors together, starting with the complex conjugate pair to streamline the process and ensure the coefficients remain real. By carefully constructing each factor from its corresponding zero, we lay the groundwork for the subsequent expansion and simplification steps, ultimately leading to the desired polynomial function.
Multiplying the Factors
The core of finding the polynomial lies in meticulously multiplying the factors we've constructed. This process involves careful algebraic manipulation, particularly when dealing with complex numbers. A strategic approach is to first multiply the complex conjugate factors (x - 1 - i) and (x - 1 + i). This pairing simplifies the process and ensures the immediate elimination of imaginary parts, maintaining real coefficients as we proceed.
Let’s begin by multiplying the complex conjugate factors:
(x - 1 - i)(x - 1 + i)
To facilitate this multiplication, we can recognize it as a difference of squares pattern by grouping (x - 1) together:
[(x - 1) - i][(x - 1) + i] = (x - 1)^2 - i^2
Expanding the square and recalling that i^2 = -1, we have:
(x^2 - 2x + 1) - (-1) = x^2 - 2x + 1 + 1 = x^2 - 2x + 2
As anticipated, the result is a quadratic expression with real coefficients. This step confirms the importance of handling complex conjugates together to satisfy the requirement of real coefficients in our polynomial. Now that we have simplified the complex portion, we move on to multiply the remaining real factors. These are (x - √6) and (x - 6). Let's multiply these together:
(x - √6)(x - 6) = x^2 - 6x - √6x + 6√6
Now, we have two expressions to multiply: (x^2 - 2x + 2) and (x^2 - 6x - √6x + 6√6). This next multiplication step is more involved, requiring us to distribute each term in the first quadratic across all terms in the second. To minimize errors, a systematic approach is crucial. We'll multiply each term of (x^2 - 2x + 2) by the entire expression (x^2 - 6x - √6x + 6√6) and then combine like terms. This meticulous process ensures that we accurately account for all terms and their coefficients. The resulting expression, though lengthy, will represent the expanded form of our desired polynomial.
Expanding and Simplifying the Polynomial
Having multiplied the initial pairs of factors, we now face the task of expanding and simplifying the resulting expressions. We have (x^2 - 2x + 2) from the complex conjugate pair and (x^2 - 6x - √6x + 6√6) from the real roots. To find the complete polynomial, we need to multiply these two expressions together.
f(x) = (x^2 - 2x + 2)(x^2 - 6x - √6x + 6√6)
This multiplication involves distributing each term in the first quadratic across all terms in the second. It's a process that requires care to avoid errors, but a systematic approach can make it manageable. Let's break it down:
*x2*(x2 - 6x - √6x + 6√6) = x^4 - 6x^3 - √6x^3 + 6√6x^2
-2x*(x^2 - 6x - √6x + 6√6) = -2x^3 + 12x^2 + 2√6x^2 - 12√6x
2*(x^2 - 6x - √6x + 6√6) = 2x^2 - 12x - 2√6x + 12√6
Now, we add these results together:
x^4 - 6x^3 - √6x^3 + 6√6x^2 - 2x^3 + 12x^2 + 2√6x^2 - 12√6x + 2x^2 - 12x - 2√6x + 12√6
The next step is to combine like terms. This involves grouping terms with the same power of x and simplifying their coefficients:
x^4 + (-6x^3 - 2x^3 - √6x^3) + (6√6x^2 + 12x^2 + 2√6x^2 + 2x^2) + (-12√6x - 2√6x - 12x) + 12√6
Combining the coefficients, we get:
x^4 - (8 + √6)x^3 + (14 + 8√6)x^2 - (12 + 14√6)x + 12√6
This is the expanded form of our polynomial. It's a quartic polynomial (degree 4) with real coefficients, as required. Each term has a real coefficient, and the polynomial is in standard form, with the powers of x descending from left to right. This expansion and simplification process demonstrates the algebraic techniques required to combine the factors and arrive at the final polynomial expression. The final step is to present the complete polynomial function, ensuring it meets the initial conditions of the problem.
Writing the Final Polynomial Function
After the intricate process of expanding and simplifying, the final step is to write the complete polynomial function. This involves compiling the results of our previous calculations into a coherent and standard polynomial form. From the previous section, we derived the polynomial:
f(x) = x^4 - (8 + √6)x^3 + (14 + 8√6)x^2 - (12 + 14√6)x + 12√6
This polynomial function, f(x), satisfies all the conditions outlined in the problem statement. It has real coefficients, meaning that each term's coefficient is a real number. It also has the specified zeros: √6, 6, 1 + i, and 1 - i. The inclusion of the complex conjugate 1 - i was crucial in ensuring the polynomial has real coefficients.
To verify this, we can conceptually substitute each zero into the polynomial and confirm that the result is zero. While the algebraic manipulation for the complex roots would be somewhat lengthy, the principle holds true. The polynomial is constructed such that these values will indeed make the function equal to zero.
It’s important to remember that this is just one possible solution. As noted earlier, we could multiply the entire polynomial by any non-zero real number and still maintain the same zeros. This means there are infinitely many polynomials that could satisfy the given conditions. For instance, if we multiplied the polynomial by 2, the resulting polynomial would have different coefficients but the same zeros. This illustrates the flexibility in constructing polynomials from a given set of zeros.
In summary, finding a polynomial function with real coefficients given specific zeros is a multi-step process. It begins with understanding the implications of complex zeros and the necessity of including their conjugates. Next, we construct the factors corresponding to each zero. These factors are then multiplied and simplified, often requiring careful algebraic manipulation. Finally, we present the complete polynomial function, ensuring it meets the initial conditions. This process combines algebraic techniques with an understanding of the properties of complex numbers, providing a robust method for constructing polynomials with specific characteristics. The result is a polynomial function that accurately reflects the given zeros and adheres to the requirement of real coefficients.
In conclusion, the polynomial function we found is:
f(x) = x^4 - (8 + √6)x^3 + (14 + 8√6)x^2 - (12 + 14√6)x + 12√6
This polynomial has real coefficients and the zeros √6, 6, and 1 + i, fulfilling the requirements of the problem.
