Finding Zeros Of Polynomial P(x) = 2x³ + 4x² - 6x A Comprehensive Guide
In mathematics, finding the zeros of a polynomial is a fundamental problem with applications in various fields, including engineering, physics, and computer science. The zeros of a polynomial, also known as roots or x-intercepts, are the values of x for which the polynomial evaluates to zero. In this comprehensive guide, we will delve into the process of finding the zeros of the polynomial p(x) = 2x³ + 4x² - 6x. We will explore various techniques, including factoring, the quadratic formula, and graphical methods, to determine the zeros of this polynomial and provide a detailed explanation of each step involved. Understanding the zeros of a polynomial is essential for analyzing its behavior and solving related mathematical problems. This article aims to provide a clear and concise explanation of how to find the zeros of the polynomial p(x), making it accessible to students, educators, and anyone interested in mathematics.
The first step in finding the zeros of the polynomial p(x) = 2x³ + 4x² - 6x is to factor it. Factoring involves expressing the polynomial as a product of simpler expressions, which makes it easier to identify the values of x that make the polynomial equal to zero. By factoring the polynomial, we can break it down into simpler components, allowing us to analyze each component separately and find the corresponding zeros. This approach simplifies the process of finding the roots and provides a clear understanding of the polynomial's structure. Factoring is a crucial technique in algebra and calculus, as it allows us to solve equations, simplify expressions, and analyze functions. By mastering factoring techniques, we can effectively solve a wide range of mathematical problems and gain a deeper understanding of polynomial behavior.
To factor the given polynomial, we can first look for a common factor among all the terms. In this case, we observe that 2x is a common factor in all the terms of the polynomial: 2x³, 4x², and -6x. Factoring out 2x from the polynomial, we get:
p(x) = 2x(x² + 2x - 3)
Now, we need to factor the quadratic expression inside the parentheses, which is x² + 2x - 3. This quadratic expression can be factored into two binomials. To factor a quadratic expression of the form ax² + bx + c, we need to find two numbers that multiply to c and add up to b. In this case, we need to find two numbers that multiply to -3 and add up to 2. The numbers 3 and -1 satisfy these conditions, since 3 * (-1) = -3 and 3 + (-1) = 2. Therefore, we can factor the quadratic expression as follows:
x² + 2x - 3 = (x + 3)(x - 1)
Substituting this back into the expression for p(x), we get:
p(x) = 2x(x + 3)(x - 1)
Now that we have factored the polynomial completely, we can easily identify its zeros. The zeros of a polynomial are the values of x that make the polynomial equal to zero. To find the zeros, we set each factor equal to zero and solve for x.
To find the zeros of the polynomial p(x) = 2x(x + 3)(x - 1), we set each factor equal to zero and solve for x. This gives us the following equations:
- 2x = 0
- x + 3 = 0
- x - 1 = 0
Solving the first equation, 2x = 0, for x, we divide both sides by 2:
x = 0
So, one of the zeros of the polynomial is x = 0. This means that the polynomial p(x) intersects the x-axis at the point (0, 0). The zero x = 0 corresponds to the factor 2x in the factored form of the polynomial. This factor indicates that the polynomial has a root at the origin, which is a crucial piece of information for understanding the polynomial's behavior near the origin. Identifying this zero is essential for sketching the graph of the polynomial and analyzing its properties.
Solving the second equation, x + 3 = 0, for x, we subtract 3 from both sides:
x = -3
So, another zero of the polynomial is x = -3. This means that the polynomial p(x) intersects the x-axis at the point (-3, 0). The zero x = -3 corresponds to the factor (x + 3) in the factored form of the polynomial. This factor indicates that the polynomial has a root at x = -3, which is an important point to consider when analyzing the polynomial's graph and behavior.
Solving the third equation, x - 1 = 0, for x, we add 1 to both sides:
x = 1
So, the third zero of the polynomial is x = 1. This means that the polynomial p(x) intersects the x-axis at the point (1, 0). The zero x = 1 corresponds to the factor (x - 1) in the factored form of the polynomial. This factor indicates that the polynomial has a root at x = 1, which is another crucial point for understanding the polynomial's graph and properties.
Therefore, the zeros of the polynomial p(x) = 2x³ + 4x² - 6x are x = 0, x = -3, and x = 1. These zeros represent the points where the graph of the polynomial intersects the x-axis. By finding these zeros, we gain valuable insights into the behavior of the polynomial and its graphical representation.
To visualize the zeros of the polynomial p(x) = 2x³ + 4x² - 6x, we can plot them on a coordinate plane. The zeros are the x-coordinates of the points where the graph of the polynomial intersects the x-axis. We have found that the zeros of the polynomial are x = 0, x = -3, and x = 1. These values represent the points where the polynomial function p(x) equals zero. Plotting these zeros on the x-axis helps us understand the behavior of the polynomial function and its relationship with the x-axis. Visualizing the zeros is an essential step in analyzing polynomial functions and their graphs.
To plot these zeros, we draw a coordinate plane with the x-axis and y-axis. Then, we mark the points x = 0, x = -3, and x = 1 on the x-axis. These points represent the x-intercepts of the graph of the polynomial. The y-coordinate of each of these points is 0, since the polynomial evaluates to zero at these values of x. Plotting the zeros on the coordinate plane provides a visual representation of the points where the polynomial function crosses or touches the x-axis. This visual representation is helpful in understanding the overall shape and behavior of the polynomial graph.
In addition to plotting the zeros, we can also sketch the general shape of the polynomial by considering its degree and leading coefficient. The degree of the polynomial p(x) = 2x³ + 4x² - 6x is 3, which is an odd number. This indicates that the graph of the polynomial will have opposite end behaviors, meaning that it will extend to positive infinity in one direction and negative infinity in the other direction. The leading coefficient of the polynomial is 2, which is positive. This indicates that the graph of the polynomial will rise to the right. By considering the degree and leading coefficient, we can get a general idea of the polynomial's shape and behavior.
Since the polynomial has three distinct real zeros, the graph of the polynomial will cross the x-axis at these three points. We can sketch the graph by starting from the left, where the graph will extend to negative infinity. As we move towards the right, the graph will cross the x-axis at x = -3. Then, it will turn around and cross the x-axis again at x = 0. Finally, it will turn around again and cross the x-axis at x = 1, and then extend to positive infinity. This sketch provides a visual representation of the polynomial's behavior and its relationship with the x-axis.
In this comprehensive guide, we have successfully found the zeros of the polynomial p(x) = 2x³ + 4x² - 6x. We began by factoring the polynomial, which allowed us to express it as a product of simpler expressions. By factoring out the common factor 2x and then factoring the resulting quadratic expression, we obtained the factored form p(x) = 2x(x + 3)(x - 1). This factored form made it easy to identify the zeros of the polynomial. Understanding the process of factoring polynomials is crucial for solving a wide range of mathematical problems.
Next, we set each factor equal to zero and solved for x to find the zeros of the polynomial. We found that the zeros are x = 0, x = -3, and x = 1. These zeros represent the points where the graph of the polynomial intersects the x-axis. Knowing the zeros of a polynomial is essential for understanding its behavior and sketching its graph. The zeros provide key information about the polynomial's roots and its relationship with the x-axis.
Finally, we discussed how to plot the zeros on a coordinate plane to visualize them. Plotting the zeros helps us understand the graphical representation of the polynomial and its relationship with the x-axis. We also discussed how to sketch the general shape of the polynomial by considering its degree and leading coefficient. The degree of the polynomial tells us about its end behavior, while the leading coefficient tells us about its direction. By combining the zeros and the general shape, we can create a complete graph of the polynomial.
In conclusion, finding the zeros of a polynomial is a fundamental problem in mathematics with various applications. By mastering the techniques of factoring, solving equations, and plotting points, we can effectively find the zeros of any polynomial and understand its behavior. The zeros of a polynomial are essential for analyzing its properties, sketching its graph, and solving related mathematical problems. This guide has provided a clear and concise explanation of how to find the zeros of the polynomial p(x) = 2x³ + 4x² - 6x, making it accessible to students, educators, and anyone interested in mathematics. We encourage readers to practice these techniques and apply them to other polynomials to further enhance their understanding.
