X-Axis Intercepts Solving Polynomial Equations A Comprehensive Guide
In mathematics, particularly in algebra, identifying the points where a graph intersects the x-axis is a fundamental concept. These points, known as x-intercepts, are crucial for understanding the behavior and characteristics of functions, especially polynomial functions. This article delves into the process of finding x-intercepts, focusing on a specific example to illustrate the key steps and concepts involved. We will explore how to determine the ordered pairs that represent these intercepts, providing a comprehensive guide for students and enthusiasts alike.
Understanding X-Axis Intercepts
X-axis intercepts, also known as roots or zeros of a function, are the points where the graph of the function crosses the x-axis. At these points, the y-coordinate is always zero. Finding these intercepts is essential for analyzing the function's behavior, such as its domain, range, and intervals of increase and decrease. For polynomial functions, the x-intercepts provide valuable information about the solutions to the equation f(x) = 0.
To find the x-intercepts of a function, we set the function equal to zero and solve for x. This process involves algebraic techniques such as factoring, using the quadratic formula, or employing numerical methods. The solutions obtained represent the x-coordinates of the points where the graph intersects the x-axis. Each x-intercept corresponds to an ordered pair of the form (x, 0), where x is the solution to the equation f(x) = 0.
The significance of x-intercepts extends beyond mere points on a graph. They provide insights into the function's real-world applications, such as determining the equilibrium points in economic models or the points of stability in physical systems. Understanding how to find and interpret x-intercepts is a cornerstone of mathematical analysis and problem-solving.
Problem: Identifying X-Axis Intercepts of a Polynomial Function
Let's consider the polynomial function f(x) = (x - 1)(x^2 + x - 20). Our goal is to determine which ordered pair represents a point where the graph of this function crosses the x-axis. This involves finding the x-intercepts of the function, which are the points where the graph intersects the x-axis. As mentioned earlier, at these points, the y-coordinate is zero.
To find the x-intercepts, we need to solve the equation f(x) = 0. This means setting the polynomial function equal to zero and finding the values of x that satisfy the equation. In this case, we have the equation (x - 1)(x^2 + x - 20) = 0. This equation is already partially factored, which simplifies the process of finding the solutions.
Step 1: Factor the Polynomial
The given polynomial function is f(x) = (x - 1)(x^2 + x - 20). To find the x-intercepts, we need to factor the quadratic expression (x^2 + x - 20). We are looking for two numbers that multiply to -20 and add to 1. These numbers are 5 and -4. Therefore, we can factor the quadratic expression as (x + 5)(x - 4).
Now, the factored form of the polynomial function is f(x) = (x - 1)(x + 5)(x - 4). This factored form is crucial for identifying the x-intercepts, as it allows us to easily determine the values of x that make the function equal to zero.
Step 2: Set Each Factor to Zero
To find the x-intercepts, we set each factor of the polynomial equal to zero and solve for x. This is based on the zero-product property, which states that if the product of several factors is zero, then at least one of the factors must be zero.
We have three factors: (x - 1), (x + 5), and (x - 4). Setting each factor to zero gives us the following equations:
- x - 1 = 0
- x + 5 = 0
- x - 4 = 0
Solving each of these equations will give us the x-coordinates of the x-intercepts.
Step 3: Solve for x
Now, we solve each equation for x:
- x - 1 = 0 => x = 1
- x + 5 = 0 => x = -5
- x - 4 = 0 => x = 4
These values of x represent the x-coordinates of the points where the graph of the function crosses the x-axis. Therefore, the x-intercepts are x = 1, x = -5, and x = 4.
Step 4: Form Ordered Pairs
Since the x-intercepts are the points where the graph crosses the x-axis, the y-coordinate at these points is always 0. Therefore, the ordered pairs representing the x-intercepts are (1, 0), (-5, 0), and (4, 0).
Analyzing the Options
Now, let's compare the ordered pairs we found with the options provided in the problem:
A. (-5, 0) B. (-1, 0) C. (20, 0) D. (-4, 0)
We can see that the ordered pair (-5, 0) matches one of the x-intercepts we found. The other options do not match our calculated x-intercepts.
Conclusion: The Correct Answer
Based on our analysis, the correct answer is A. (-5, 0). This ordered pair represents a point where the graph of the function f(x) = (x - 1)(x^2 + x - 20) crosses the x-axis.
Finding the x-intercepts of a polynomial function is a crucial skill in algebra. By factoring the polynomial, setting each factor to zero, and solving for x, we can determine the points where the graph intersects the x-axis. These x-intercepts provide valuable information about the function's behavior and solutions.
In this article, we have demonstrated a step-by-step approach to solving this type of problem. By understanding the concepts and techniques involved, students can confidently tackle similar problems and deepen their understanding of polynomial functions.
This detailed explanation not only provides the correct answer but also reinforces the underlying mathematical principles, making it a valuable resource for learning and review.
