Polynomial Function With Leading Coefficient 1 And Roots (7+i) And (5-i)
Understanding Polynomial Functions and Roots
When delving into the world of polynomial functions, understanding the concept of roots is paramount. Roots, also known as zeros, are the values of x for which the polynomial function equals zero. These roots provide valuable information about the function's behavior and its graph. Specifically, when dealing with complex roots, the situation becomes even more interesting due to the properties of complex numbers and their conjugates. In our main question, we aim to identify a polynomial function given its roots, a leading coefficient, and the multiplicity of those roots. This requires a solid grasp of how roots relate to factors of a polynomial and how complex conjugates play a role in polynomial construction. Complex roots, a critical concept in polynomial functions, always come in conjugate pairs if the polynomial has real coefficients. This means that if (a + bi) is a root, then (a - bi) must also be a root. This property is crucial when constructing polynomials with complex roots. In our specific problem, we're given the roots (7 + i) and (5 - i), which are complex numbers. Therefore, their conjugates, (7 - i) and (5 + i), must also be roots of the polynomial. The multiplicity of a root indicates how many times that root appears as a solution of the polynomial equation. A root with multiplicity 1 means it appears only once. Understanding the concept of multiplicity is crucial for accurately constructing the polynomial function, ensuring it has the correct degree and behavior near its roots. Now, let's consider how roots translate into factors of a polynomial. If r is a root of a polynomial function, then (x - r) is a factor of that polynomial. This relationship is the cornerstone of building a polynomial from its roots. By understanding this connection, we can reverse-engineer the polynomial function from its given roots. For each root, we create a corresponding factor, and by multiplying these factors together, we can construct the polynomial. Given that the leading coefficient is 1, we can ensure that the polynomial expands to have a leading term with a coefficient of 1, which simplifies the process of determining the correct polynomial function. This leading coefficient plays a significant role in determining the overall shape and behavior of the polynomial graph, especially as x approaches positive or negative infinity. Therefore, keeping this coefficient in mind is essential for constructing the correct polynomial. In the subsequent sections, we will apply these concepts to construct the polynomial function that satisfies the given conditions. We'll start by identifying all the roots, including the complex conjugates, then form the corresponding factors, and finally multiply these factors together to obtain the polynomial function. This step-by-step approach will ensure that we arrive at the correct answer while reinforcing the fundamental principles of polynomial construction.
Constructing the Polynomial Function from Roots
To construct the polynomial function, we start with the given roots: (7 + i) and (5 - i), each with a multiplicity of 1. Remember, complex roots of polynomials with real coefficients always come in conjugate pairs. Therefore, if (7 + i) is a root, its conjugate (7 - i) must also be a root. Similarly, if (5 - i) is a root, its conjugate (5 + i) must also be a root. This gives us a complete set of roots: (7 + i), (7 - i), (5 - i), and (5 + i). Knowing these roots is the first step in constructing our polynomial, and understanding the conjugate pairs is crucial for accuracy. Each root corresponds to a factor of the polynomial. If r is a root, then (x - r) is a factor. So, for the roots we have, the factors are: [x - (7 + i)], [x - (7 - i)], [x - (5 - i)], and [x - (5 + i)]. These factors are the building blocks of our polynomial, and by multiplying them together, we will obtain the polynomial function. Now, we multiply these factors together. A strategic approach is to first multiply the factors corresponding to conjugate pairs, as this will simplify the calculations by eliminating the imaginary terms early on. Let's multiply [x - (7 + i)] and [x - (7 - i)]:
[ (x - (7 + i)) * (x - (7 - i)) = (x - 7 - i) * (x - 7 + i) ]
Expanding this product, we get:
[ x^2 - 7x + ix - 7x + 49 - 7i - ix + 7i - i^2 ]
Since i^2 = -1, we simplify the expression:
[ x^2 - 14x + 49 - (-1) = x^2 - 14x + 50 ]
Next, we multiply [x - (5 - i)] and [x - (5 + i)]:
[ (x - (5 - i)) * (x - (5 + i)) = (x - 5 + i) * (x - 5 - i) ]
Expanding this product, we get:
[ x^2 - 5x - ix - 5x + 25 + 5i + ix - 5i - i^2 ]
Again, since i^2 = -1, we simplify the expression:
[ x^2 - 10x + 25 - (-1) = x^2 - 10x + 26 ]
Now, we multiply the two resulting quadratic expressions:
[ (x^2 - 14x + 50) * (x^2 - 10x + 26) ]
This multiplication gives us:
[ x^4 - 10x^3 + 26x^2 - 14x^3 + 140x^2 - 364x + 50x^2 - 500x + 1300 ]
Combining like terms, we obtain the polynomial function:
[ f(x) = x^4 - 24x^3 + 216x^2 - 864x + 1300 ]
This polynomial function has a leading coefficient of 1 and roots (7 + i), (7 - i), (5 - i), and (5 + i), each with multiplicity 1. This completes the construction process, providing us with the polynomial function that meets all the given criteria.
Analyzing the Answer Choices
After constructing the polynomial function from the given roots, we need to analyze the answer choices to identify the correct one. The constructed polynomial is [ f(x) = x^4 - 24x^3 + 216x^2 - 864x + 1300 ]. We will compare this with the provided options, focusing on how each option represents the polynomial in factored form. Remember that each root corresponds to a factor of the polynomial. The roots are (7 + i), (7 - i), (5 - i), and (5 + i), so the factors should be [x - (7 + i)], [x - (7 - i)], [x - (5 - i)], and [x - (5 + i)]. Now, let's examine the given answer choices:
A. [ f(x) = (x + 7)(x - i)(x + 5)(x + i) ]
This option does not correctly represent the factors corresponding to the given roots. The factors do not account for the complex conjugates, and the signs are incorrect for the real parts of the roots. Therefore, this option is incorrect.
B. [ f(x) = (x - 7)(x - i)(x - 5)(x + i) ]
Similarly, this option also fails to accurately represent the factors. It doesn't account for the conjugate pairs and has incorrect signs for the roots. Thus, this option is also incorrect.
C. [ f(x) = (x - (7 - i))(x - (5 + i))(x - (7 + i))(x - (5 - i)) ]
This option correctly represents the factors corresponding to the roots (7 + i), (7 - i), (5 - i), and (5 + i). Each factor is in the form (x - r), where r is a root of the polynomial. This aligns perfectly with our understanding of how roots relate to factors of a polynomial. Therefore, this option is the correct representation of the polynomial function in factored form.
To further solidify our understanding, let's expand the factors in option C to see if it matches the polynomial we constructed earlier:
[ (x - (7 - i))(x - (5 + i))(x - (7 + i))(x - (5 - i)) ]
Expanding these factors, we would indeed arrive at the polynomial [ f(x) = x^4 - 24x^3 + 216x^2 - 864x + 1300 ], confirming that option C is the correct answer. This process of comparing and expanding reinforces the relationship between roots, factors, and the polynomial function itself.
Conclusion: Identifying the Correct Polynomial Function
In conclusion, the polynomial function with a leading coefficient of 1 and roots (7 + i) and (5 - i), each with multiplicity 1, is represented by option C: [ f(x) = (x - (7 - i))(x - (5 + i))(x - (7 + i))(x - (5 - i)) ]. Our journey to identify the correct polynomial involved several key steps. First, we understood the fundamental relationship between roots and factors of a polynomial. We recognized that complex roots come in conjugate pairs, ensuring that if (7 + i) and (5 - i) are roots, then (7 - i) and (5 + i) must also be roots. This understanding allowed us to form the correct factors for the polynomial. Next, we strategically multiplied the factors, starting with the conjugate pairs, to simplify the calculations. This step-by-step approach enabled us to arrive at the polynomial in standard form: [ f(x) = x^4 - 24x^3 + 216x^2 - 864x + 1300 ]. Finally, we analyzed the answer choices, comparing them to the factors we had derived from the roots. Option C stood out as the correct representation of the polynomial in factored form. We further validated our answer by expanding the factors in option C, confirming that it matches the polynomial we constructed. This comprehensive approach not only led us to the correct answer but also reinforced our understanding of polynomial functions, roots, and factors. The process highlights the importance of understanding complex conjugates, the relationship between roots and factors, and strategic multiplication techniques. By mastering these concepts, one can confidently tackle similar problems involving polynomial construction. This problem serves as a valuable exercise in applying the principles of polynomial algebra and reinforces the significance of accuracy and attention to detail in mathematical problem-solving. The ability to construct polynomials from their roots is a fundamental skill in algebra and calculus, with applications in various fields such as engineering, physics, and computer science. Therefore, a thorough understanding of this topic is crucial for success in higher-level mathematics and its applications.
