Zeros Of Polynomial Functions A Detailed Analysis Of F(x) = X³ + 5x² + X + 5

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In the realm of mathematics, particularly in algebra, understanding the nature and number of zeros of a polynomial function is a fundamental concept. Zeros, also known as roots, are the values of x for which the function f(x) equals zero. These zeros provide critical insights into the behavior and graph of the polynomial. In this comprehensive guide, we will delve into the process of determining the number of complex and real zeros for a given polynomial function, using the example of f(x) = x³ + 5x² + x + 5. This exploration will involve applying key theorems and techniques in algebra, such as the Fundamental Theorem of Algebra, the Rational Root Theorem, and factorization methods. By the end of this discussion, you will have a clear understanding of how to analyze polynomial functions and identify their zeros effectively.

Fundamental Concepts: Real vs. Complex Zeros

Before we dive into the specifics of the given function, let's clarify the distinction between real and complex zeros. A real zero is a root of the polynomial that is a real number. This means it can be plotted on the number line. These zeros correspond to the points where the graph of the function intersects the x-axis. On the other hand, a complex zero is a root that involves the imaginary unit i, where i is defined as the square root of -1. Complex zeros come in conjugate pairs for polynomials with real coefficients, meaning if a + bi is a zero, then a - bi is also a zero. Understanding this distinction is crucial because it directly impacts how we interpret the solutions of polynomial equations and their graphical representations. Real zeros provide tangible intersection points on the graph, while complex zeros, though not directly visible on the real number plane, play a significant role in the overall algebraic structure and behavior of the polynomial.

Applying the Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra is a cornerstone in determining the number of zeros a polynomial function possesses. This theorem states that a polynomial of degree n, where n is a positive integer, has exactly n complex zeros, counting multiplicities. Multiplicity refers to the number of times a particular zero appears as a root of the polynomial. For instance, if a factor (x - a) appears twice in the factored form of the polynomial, then a is a zero with a multiplicity of 2. In the case of our function, f(x) = x³ + 5x² + x + 5, the degree of the polynomial is 3. Therefore, according to the Fundamental Theorem of Algebra, this function has exactly 3 complex zeros. This theorem assures us of the total number of zeros, but it does not specify how many of these zeros are real and how many are non-real (complex with an imaginary component). To determine the nature of these zeros, we need to employ further techniques such as factorization and the Rational Root Theorem.

Factoring the Polynomial

Factoring the polynomial is a powerful method to find its zeros. By expressing the polynomial as a product of simpler factors, we can easily identify the values of x that make the polynomial equal to zero. For the given function, f(x) = x³ + 5x² + x + 5, we can use a technique called factoring by grouping. This involves grouping terms in pairs and factoring out common factors. Let's apply this to our function:

  1. Group the terms: (x³ + 5x²) + (x + 5)
  2. Factor out the common factor from each group: x²(x + 5) + 1(x + 5)
  3. Notice that (x + 5) is a common factor in both terms. Factor it out: (x + 5)(x² + 1)

Now we have factored the polynomial into (x + 5)(x² + 1). This factorization is a crucial step because it allows us to identify the zeros more easily. The zeros are the values of x that make each factor equal to zero. So, we set each factor equal to zero and solve for x.

Identifying Real Zeros

To identify the real zeros, we focus on the factors that will yield real number solutions. From the factored form (x + 5)(x² + 1), the first factor, (x + 5), gives us a real zero. Setting this factor equal to zero, we have:

  • x + 5 = 0
  • x = -5

So, x = -5 is a real zero of the polynomial f(x) = x³ + 5x² + x + 5. This means that the graph of the function will intersect the x-axis at the point (-5, 0). Now, we need to consider the second factor, (x² + 1), to determine if it yields any other real zeros or complex zeros. Understanding how to extract real zeros from the factored form is essential, as it directly relates to the points where the polynomial's graph crosses the x-axis, providing a visual representation of the function's behavior.

Determining Complex Zeros

Next, we investigate the factor (x² + 1) to find any additional zeros. Setting this factor equal to zero gives us:

  • x² + 1 = 0
  • x² = -1
  • x = ±√(-1)
  • x = ±i

Here, we find that the solutions are x = i and x = -i, where i is the imaginary unit (√(-1)). These are complex zeros because they involve the imaginary unit. Since complex zeros occur in conjugate pairs for polynomials with real coefficients, the presence of i implies that -i is also a zero. This is consistent with our result. The complex zeros do not correspond to x-intercepts on the graph of the polynomial, but they are crucial in understanding the complete set of solutions for the polynomial equation.

Counting Real and Complex Zeros

Now that we have found all the zeros of the function f(x) = x³ + 5x² + x + 5, let's count how many are real and how many are complex. We found one real zero, x = -5, and two complex zeros, x = i and x = -i. This gives us a total of three zeros, which aligns with the Fundamental Theorem of Algebra, which states that a polynomial of degree 3 has 3 complex zeros, counting multiplicities. In this case, each zero has a multiplicity of 1. Therefore, the function f(x) = x³ + 5x² + x + 5 has:

  • 1 real zero: x = -5
  • 2 complex zeros: x = i and x = -i

This distribution of zeros tells us a lot about the behavior of the polynomial function. The single real zero indicates that the graph of the function crosses the x-axis only once. The pair of complex zeros means that the graph does not have any other x-intercepts. Understanding the number and nature of zeros is fundamental for sketching the graph of a polynomial and analyzing its properties.

Conclusion

In summary, by applying the Fundamental Theorem of Algebra, factoring techniques, and solving for zeros, we have determined that the function f(x) = x³ + 5x² + x + 5 has 3 complex zeros, with 1 being real and 2 being non-real (complex). Specifically, the real zero is x = -5, and the complex zeros are x = i and x = -i. This analysis underscores the importance of understanding fundamental algebraic principles in characterizing polynomial functions. The ability to identify and classify zeros allows for a deeper understanding of the function's behavior, its graph, and its solutions. This knowledge is not only crucial in academic mathematics but also has practical applications in various fields, including engineering, physics, and computer science. By mastering these techniques, one can confidently approach the analysis of polynomial functions and their properties.

Therefore, the correct answer is:

D. 3 complex zeros; 1 real