Finding Roots Of Polynomial Function F(x) = (x^2 + 4)(x - 2) Real And Non-Real Solutions

by Admin 89 views

Introduction: Understanding Roots and Polynomial Functions

In the realm of mathematics, understanding the roots of a function is a fundamental concept. The roots, also known as zeros or solutions, are the values of x for which the function f(x) equals zero. Finding these roots is crucial in various applications, including engineering, physics, and computer science. In this article, we will delve into the process of finding both real and non-real roots of the given function $f(x) = (x^2 + 4)(x - 2)$. Polynomial functions, like the one we are examining, play a significant role in modeling real-world phenomena. Their roots provide valuable insights into the behavior and characteristics of these models. The roots of a polynomial equation are the values of the variable that make the equation true. In other words, they are the points where the graph of the polynomial intersects the x-axis. For a polynomial of degree n, there are n roots, counting multiplicity, in the complex number system. These roots can be real numbers or complex numbers. Complex numbers have a real part and an imaginary part, while real numbers have only a real part. The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This theorem is the cornerstone of understanding the number and nature of roots for polynomial functions. Therefore, to master the art of mathematical analysis, a comprehensive understanding of polynomial roots, both real and non-real, is indispensable. This article will serve as a guide to navigate this critical area of mathematical analysis, ensuring clarity and precision in every step. By dissecting the function $f(x) = (x^2 + 4)(x - 2)$, we aim to provide a clear methodology applicable to a broader range of polynomial equations.

Deconstructing the Function: f(x) = (x^2 + 4)(x - 2)

The first step in finding the roots of $f(x) = (x^2 + 4)(x - 2)$ is to understand its structure. This function is a product of two factors: a quadratic factor $(x^2 + 4)$ and a linear factor $(x - 2)$. To find the roots, we set the function equal to zero: $(x^2 + 4)(x - 2) = 0$. This equation holds true if either of the factors is equal to zero. Therefore, we can split the problem into two simpler equations: $x^2 + 4 = 0$ and $x - 2 = 0$. Analyzing the linear factor first, $x - 2 = 0$ is straightforward. Adding 2 to both sides, we find one of the roots: $x = 2$. This is a real root, meaning it is a real number. Now, let's turn our attention to the quadratic factor, $x^2 + 4 = 0$. This equation requires a bit more manipulation. Subtracting 4 from both sides, we get $x^2 = -4$. To find x, we need to take the square root of both sides: $x = \\pm \\sqrt{-4}$. Since we are dealing with the square root of a negative number, we will encounter imaginary numbers. Recall that the imaginary unit i is defined as $i = \\sqrt{-1}$. Therefore, we can rewrite $x = \\pm \\sqrt{-4}$ as $x = \\pm \\sqrt{4 \\cdot -1} = \\pm \\sqrt{4} \\cdot \\sqrt{-1} = \\pm 2i$. These roots, $x = 2i$ and $x = -2i$, are non-real roots because they involve the imaginary unit i. In summary, by deconstructing the function into its factors and setting each factor to zero, we have successfully identified all the roots. The linear factor gave us a real root, while the quadratic factor yielded two non-real roots. This method of factorization is a powerful technique for finding roots of polynomial functions, allowing us to systematically break down complex equations into simpler components.

Solving for Roots: Real and Non-Real Solutions

Now, let's solve each equation to find the roots. For the linear factor $(x - 2) = 0$, solving for x is simple. Add 2 to both sides of the equation: $x = 2$. This gives us our first root, which is a real root. This means that the function f(x) intersects the x-axis at x = 2. Next, we tackle the quadratic factor $(x^2 + 4) = 0$. To solve for x, we first subtract 4 from both sides: $x^2 = -4$. Now, we take the square root of both sides: $x = \\pm \\sqrt-4}$. Since we have a negative number under the square root, we introduce the imaginary unit i, where $i = \\sqrt{-1}$. We can rewrite the equation as $x = \\pm \\sqrt{4 \\cdot -1 = \\pm \\sqrt{4} \\cdot \\sqrt{-1} = \\pm 2i$. This gives us two non-real roots: $x = 2i$ and $x = -2i$. These roots are complex numbers, meaning they have both a real and an imaginary part (in this case, the real part is 0). Non-real roots do not correspond to x-intercepts on the graph of the function. Therefore, by systematically solving each factor of the function, we have found all the roots. We have one real root, x = 2, and two non-real roots, x = 2i and x = -2i. These roots completely describe the solutions to the equation $f(x) = 0$. Understanding the nature of these roots is crucial for a complete analysis of the function's behavior.

Identifying the Roots: Real Root and Complex Conjugate Pair

After solving the equations derived from the factors of $f(x)$, we can now explicitly identify the roots. From the linear factor $(x - 2)$, we found the root $x = 2$. This is a real root, as it is a real number. Real roots correspond to the points where the graph of the function intersects the x-axis. In this case, the graph of $f(x)$ crosses the x-axis at $x = 2$. From the quadratic factor $(x^2 + 4)$, we found two roots: $x = 2i$ and $x = -2i$. These are non-real roots, also known as complex roots, because they involve the imaginary unit i. Non-real roots do not appear as x-intercepts on the graph of the function. An important observation here is that the non-real roots are a complex conjugate pair. Complex conjugate pairs occur when a quadratic equation with real coefficients has non-real roots. The roots are of the form $a + bi$ and $a - bi$, where a and b are real numbers and i is the imaginary unit. In our case, the roots are $0 + 2i$ and $0 - 2i$, which clearly fit the form of a complex conjugate pair. The presence of a complex conjugate pair is a characteristic feature of polynomial functions with real coefficients. It indicates that the imaginary parts of the roots are equal in magnitude but opposite in sign. Therefore, by identifying the roots as one real root and a complex conjugate pair, we gain a deeper understanding of the function's behavior and its relationship to the complex number system. This complete identification of the roots is essential for further analysis and applications of the function.

Conclusion: The Roots of f(x) and Their Significance

In conclusion, we have successfully found all the roots of the function $f(x) = (x^2 + 4)(x - 2)$. We identified one real root, $x = 2$, and two non-real roots, $x = 2i$ and $x = -2i$. The non-real roots form a complex conjugate pair, a characteristic feature of polynomial functions with real coefficients. These roots provide valuable information about the behavior of the function. The real root, $x = 2$, indicates where the graph of the function intersects the x-axis. The non-real roots, while not visible on the real number line, play a crucial role in the function's overall shape and properties. Understanding the roots of a function is fundamental in various mathematical and scientific applications. It allows us to solve equations, model real-world phenomena, and analyze the behavior of systems. In this specific example, the roots tell us about the function's intercepts and its oscillatory behavior in the complex plane. The method we used to find the roots – factoring the function and setting each factor to zero – is a powerful technique applicable to a wide range of polynomial functions. By mastering this technique, you can confidently tackle more complex problems and gain a deeper appreciation for the beauty and power of mathematics. The significance of finding all roots, both real and non-real, cannot be overstated. It provides a complete picture of the function's solutions and allows for accurate predictions and interpretations in various contexts. Therefore, the process of finding and classifying roots is a cornerstone of mathematical analysis and problem-solving.

Final Answer

The roots of the function $f(x) = (x^2 + 4)(x - 2)$ are:

  • x = 2$ (real root)

  • x = 2i$ (non-real root)

  • x = -2i$ (non-real root)

Therefore, the correct answer is A) $x= \\pm 2 i, 2$.