X-Intercepts Of F(x) = X^4 - 5x^2 A Comprehensive Guide
In the realm of mathematics, particularly when dealing with polynomial functions, x-intercepts play a crucial role in understanding the behavior and characteristics of a graph. X-intercepts, also known as roots or zeros of a function, are the points where the graph of the function intersects the x-axis. At these points, the value of the function, denoted as f(x), is equal to zero. Finding the x-intercepts of a polynomial function is a fundamental task in algebra and calculus, as it provides valuable information about the function's solutions and its overall shape. To effectively determine the number of x-intercepts, we need to analyze the given polynomial function and employ various algebraic techniques. In this article, we will delve into the process of identifying x-intercepts, focusing on the specific polynomial function f(x) = x^4 - 5x^2. By understanding how to find x-intercepts, we gain a deeper insight into the nature of polynomial functions and their graphical representations. The significance of x-intercepts extends beyond mere points on a graph; they provide crucial information about the function's behavior, such as where the function changes sign and where it has potential maximum or minimum values. This knowledge is essential in various applications, including optimization problems, curve sketching, and real-world modeling. Furthermore, understanding x-intercepts is a stepping stone to more advanced concepts in mathematics, such as finding the roots of equations and analyzing the stability of systems. Therefore, mastering the techniques for finding x-intercepts is a valuable skill for anyone pursuing studies in mathematics, science, or engineering.
To determine the number of x-intercepts for the polynomial function f(x) = x^4 - 5x^2, we need to find the values of x for which f(x) = 0. This involves solving the equation x^4 - 5x^2 = 0. The first step in solving this equation is to factor out the common factor, which in this case is x^2. Factoring x^2 from both terms, we get x2(x2 - 5) = 0. Now, we have a product of two factors that equals zero, which means that at least one of the factors must be zero. This leads us to two separate equations: x^2 = 0 and x^2 - 5 = 0. Solving the first equation, x^2 = 0, we find that x = 0 is a solution. However, since the exponent is 2, this solution has a multiplicity of 2, meaning it is a repeated root. This implies that the graph of the function touches the x-axis at x = 0 but does not cross it. Moving on to the second equation, x^2 - 5 = 0, we need to isolate x^2. Adding 5 to both sides of the equation, we get x^2 = 5. To solve for x, we take the square root of both sides, remembering to consider both the positive and negative square roots. This gives us x = â5 and x = -â5 as the other two solutions. These are distinct real roots, indicating that the graph of the function crosses the x-axis at these points. In summary, we have found three solutions for the equation x^4 - 5x^2 = 0: x = 0 (with multiplicity 2), x = â5, and x = -â5. Each of these solutions corresponds to an x-intercept on the graph of the function. By analyzing the polynomial function and solving for its roots, we have successfully identified the x-intercepts and gained valuable insights into the function's behavior. This process demonstrates the importance of algebraic techniques in understanding the graphical representation of polynomial functions.
Finding the x-intercepts of a polynomial function involves setting the function equal to zero and solving for the variable x. In the given function, f(x) = x^4 - 5x^2, we set f(x) = 0, resulting in the equation x^4 - 5x^2 = 0. The next step is to factor the equation to simplify it and make it easier to solve. As we observed earlier, both terms in the equation have a common factor of x^2. Factoring out x^2, we get x2(x2 - 5) = 0. This factored form allows us to use the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. Applying the zero-product property to our equation, we set each factor equal to zero: x^2 = 0 and x^2 - 5 = 0. Solving the first equation, x^2 = 0, we find that x = 0. This is one x-intercept. However, since the factor x^2 appears, this root has a multiplicity of 2, meaning it is a repeated root. This indicates that the graph of the function touches the x-axis at x = 0 but does not cross it. Moving on to the second equation, x^2 - 5 = 0, we need to isolate x^2. Adding 5 to both sides of the equation, we get x^2 = 5. To solve for x, we take the square root of both sides, remembering to consider both the positive and negative square roots. This gives us x = â5 and x = -â5 as the other two solutions. These are distinct real roots, indicating that the graph of the function crosses the x-axis at these points. Therefore, the x-intercepts of the function f(x) = x^4 - 5x^2 are x = 0, x = â5, and x = -â5. These points represent where the graph of the function intersects the x-axis. Understanding how to find x-intercepts is crucial in analyzing the behavior of polynomial functions and their graphical representations.
After finding the x-intercepts of the polynomial function f(x) = x^4 - 5x^2, we can now determine the total number of x-intercepts. From our previous analysis, we found three distinct x-intercepts: x = 0, x = â5, and x = -â5. The x-intercept at x = 0 has a multiplicity of 2, which means it is a repeated root. However, when counting the number of distinct x-intercepts, we only count it once. The other two x-intercepts, x = â5 and x = -â5, are distinct real roots. Therefore, the graph of the function f(x) = x^4 - 5x^2 intersects the x-axis at three distinct points. This means that there are three x-intercepts in total. The x-intercepts provide valuable information about the behavior of the function. The x-intercept at x = 0 indicates that the graph touches the x-axis at this point but does not cross it. This is because the root has an even multiplicity (multiplicity of 2). The x-intercepts at x = â5 and x = -â5 indicate that the graph crosses the x-axis at these points. By understanding the x-intercepts, we can gain insights into the overall shape and characteristics of the graph of the polynomial function. In summary, the polynomial function f(x) = x^4 - 5x^2 has three x-intercepts, which correspond to the points where the graph intersects the x-axis. This information is crucial in analyzing the function's behavior and its graphical representation. Understanding how to determine the number of x-intercepts is a fundamental skill in algebra and calculus, as it provides valuable insights into the nature of polynomial functions.
In conclusion, by analyzing the polynomial function f(x) = x^4 - 5x^2, we have successfully determined that it has three x-intercepts. These x-intercepts are located at x = 0, x = â5, and x = -â5. The process involved setting the function equal to zero, factoring the equation, and solving for x. We found that x = 0 is a repeated root with a multiplicity of 2, while x = â5 and x = -â5 are distinct real roots. This analysis provides valuable information about the graph of the function, indicating that it intersects the x-axis at three distinct points. Understanding how to find and interpret x-intercepts is a fundamental skill in algebra and calculus. X-intercepts are crucial in analyzing the behavior of polynomial functions and their graphical representations. They provide insights into the solutions of the equation f(x) = 0 and help us understand where the function changes sign. Furthermore, x-intercepts are essential in various applications, including optimization problems, curve sketching, and real-world modeling. By mastering the techniques for finding x-intercepts, we can gain a deeper understanding of polynomial functions and their properties. This knowledge is essential for anyone pursuing studies in mathematics, science, or engineering. Therefore, the ability to determine the number of x-intercepts and their locations is a valuable tool in the study of polynomial functions and their applications. The process of finding x-intercepts involves algebraic techniques such as factoring and solving equations, which are fundamental skills in mathematics.
Therefore, the answer is C. 3 x-intercepts
