Finding Zeros Of Polynomial P(x) = (x^2 - 1)(x^2 - 5x + 6)
In the realm of mathematics, specifically within the study of polynomials, identifying the zeros (or roots) of a polynomial function is a fundamental concept. Zeros are the values of x for which the polynomial p(x) equals zero. Finding these zeros is crucial for various applications, including graphing polynomials, solving equations, and understanding the behavior of functions. This article delves into a detailed exploration of how to find the zeros of the polynomial p(x) = (x^2 - 1)(x^2 - 5x + 6). We will break down the polynomial into its constituent factors, analyze each factor, and ultimately determine the values of x that make the polynomial equal to zero. This process involves utilizing factoring techniques, solving quadratic equations, and applying the zero-product property. Understanding these methods will not only help in finding the zeros of this specific polynomial but also provide a solid foundation for tackling more complex polynomial equations in the future. Let's embark on this mathematical journey and unravel the intricacies of finding polynomial zeros.
Understanding the Polynomial
To begin our exploration of finding the zeros of the polynomial, let's first understand the structure of the given equation: p(x) = (x^2 - 1)(x^2 - 5x + 6). This polynomial is expressed as a product of two quadratic factors. Recognizing this structure is crucial because it simplifies the process of finding the zeros. Each quadratic factor can be analyzed independently, and then the zeros of the entire polynomial can be determined by combining the zeros of each factor. The first factor, (x^2 - 1), is a difference of squares, a common pattern in algebra that can be easily factored. The second factor, (x^2 - 5x + 6), is a standard quadratic expression that can be factored by finding two numbers that multiply to 6 and add up to -5. By understanding the individual components of the polynomial, we can apply appropriate factoring techniques to simplify the expression and identify the values of x that make the polynomial equal to zero. This initial step of understanding the polynomial's structure sets the stage for the subsequent steps in our quest to find its zeros. By breaking down the complex polynomial into simpler, manageable factors, we can systematically determine the values of x that satisfy the equation p(x) = 0. This approach highlights the importance of pattern recognition and strategic simplification in solving mathematical problems. So, with a clear understanding of the polynomial's structure, let's proceed to factor each quadratic expression.
Factoring the Quadratic Expressions
The core of finding the zeros of the polynomial p(x) = (x^2 - 1)(x^2 - 5x + 6) lies in factoring the quadratic expressions. Let's begin with the first factor, (x^2 - 1). As previously mentioned, this expression is a difference of squares, which can be factored using the formula a^2 - b^2 = (a - b)(a + b). Applying this formula to our expression, we get x^2 - 1 = (x - 1)(x + 1). This factorization reveals two potential zeros: x = 1 and x = -1. Now, let's turn our attention to the second quadratic expression, (x^2 - 5x + 6). To factor this expression, we need to find two numbers that multiply to 6 (the constant term) and add up to -5 (the coefficient of the x term). These numbers are -2 and -3. Therefore, we can factor the quadratic as (x - 2)(x - 3). This factorization reveals two more potential zeros: x = 2 and x = 3. By successfully factoring both quadratic expressions, we have transformed the original polynomial into a product of four linear factors: p(x) = (x - 1)(x + 1)(x - 2)(x - 3). This factored form is crucial because it directly leads us to the zeros of the polynomial. Each factor represents a potential zero, and by setting each factor equal to zero, we can determine the values of x that make the entire polynomial equal to zero. The process of factoring not only simplifies the polynomial but also provides a clear pathway to identifying its zeros, showcasing the power of algebraic manipulation in problem-solving.
Applying the Zero-Product Property
With the polynomial p(x) = (x^2 - 1)(x^2 - 5x + 6) successfully factored into p(x) = (x - 1)(x + 1)(x - 2)(x - 3), we can now apply the zero-product property to find the zeros. The zero-product property states that if the product of several factors is equal to zero, then at least one of the factors must be equal to zero. In other words, if a * b* * c* = 0, then either a = 0, b = 0, or c = 0 (or any combination thereof). Applying this property to our factored polynomial, we set each factor equal to zero and solve for x:
- x - 1 = 0 => x = 1
- x + 1 = 0 => x = -1
- x - 2 = 0 => x = 2
- x - 3 = 0 => x = 3
This process yields four distinct values for x: 1, -1, 2, and 3. These values are the zeros of the polynomial p(x). They are the points where the graph of the polynomial intersects the x-axis. The zero-product property is a fundamental tool in algebra for solving equations involving factored expressions. It allows us to break down a complex equation into simpler equations, each of which can be solved independently. By applying this property, we have efficiently identified all the zeros of the given polynomial. The zeros provide valuable information about the behavior of the polynomial function and are essential for various mathematical applications, including graphing, solving equations, and analyzing functions. Thus, understanding and applying the zero-product property is crucial for mastering polynomial algebra.
Identifying the Zeros
Having factored the polynomial p(x) = (x^2 - 1)(x^2 - 5x + 6) and applied the zero-product property, we have successfully identified the zeros of the polynomial. The zeros are the values of x that make the polynomial equal to zero. As we found in the previous section, these values are x = 1, x = -1, x = 2, and x = 3. These zeros are crucial for understanding the behavior of the polynomial function. They represent the points where the graph of the polynomial intersects the x-axis. In other words, at these x-values, the y-value of the polynomial is zero. The zeros of a polynomial provide valuable information about its roots, its factors, and its overall shape. For instance, the number of zeros (counting multiplicity) is equal to the degree of the polynomial. In this case, the polynomial is of degree 4 (since it is a product of two quadratic factors), and we have found four distinct zeros. This confirms the fundamental theorem of algebra, which states that a polynomial of degree n has exactly n complex roots (counting multiplicity). Furthermore, the zeros allow us to write the polynomial in its factored form, which can be useful for various purposes, such as simplifying expressions, solving equations, and analyzing the function's behavior. By identifying the zeros, we have gained a comprehensive understanding of the polynomial p(x) and its properties. These zeros serve as key reference points for further analysis and applications of the polynomial function. Thus, the process of finding zeros is a fundamental step in the study of polynomials.
Graphical Interpretation of Zeros
The graphical interpretation of zeros provides a visual understanding of their significance in the context of a polynomial function. The zeros of a polynomial, as we've determined for p(x) = (x^2 - 1)(x^2 - 5x + 6), correspond to the points where the graph of the polynomial intersects the x-axis. In our case, the zeros are x = 1, x = -1, x = 2, and x = 3. This means that the graph of p(x) will cross or touch the x-axis at these four points. Visualizing these intersections helps us understand the behavior of the polynomial function. For example, between any two consecutive zeros, the graph of the polynomial will either be entirely above the x-axis (positive) or entirely below the x-axis (negative). This is because the polynomial's sign can only change at its zeros. The zeros also provide information about the factors of the polynomial. Each zero corresponds to a linear factor of the polynomial. For instance, the zero x = 1 corresponds to the factor (x - 1), x = -1 corresponds to (x + 1), x = 2 corresponds to (x - 2), and x = 3 corresponds to (x - 3). This relationship between zeros and factors is a fundamental concept in polynomial algebra. Graphing the polynomial can further enhance our understanding of its behavior. By plotting the zeros on the x-axis and considering the polynomial's degree and leading coefficient, we can sketch a general shape of the graph. The graph will oscillate around the x-axis, crossing it at the zeros. The number of turning points (local maxima and minima) in the graph is at most one less than the degree of the polynomial. Thus, the graphical interpretation of zeros provides a powerful tool for visualizing and understanding the properties of polynomial functions. It connects the algebraic concept of zeros with the geometric representation of the graph, offering a comprehensive perspective on polynomial behavior.
Conclusion
In conclusion, finding the zeros of the polynomial p(x) = (x^2 - 1)(x^2 - 5x + 6) involved a systematic approach that combined factoring techniques, the zero-product property, and graphical interpretation. We began by understanding the structure of the polynomial, recognizing it as a product of two quadratic factors. We then factored each quadratic expression, utilizing the difference of squares pattern for (x^2 - 1) and standard factoring methods for (x^2 - 5x + 6). This process transformed the polynomial into its factored form: p(x) = (x - 1)(x + 1)(x - 2)(x - 3). Next, we applied the zero-product property, setting each factor equal to zero and solving for x. This yielded the zeros of the polynomial: x = 1, x = -1, x = 2, and x = 3. These zeros represent the values of x for which the polynomial equals zero and are crucial for understanding its behavior. Finally, we explored the graphical interpretation of zeros, recognizing that they correspond to the points where the graph of the polynomial intersects the x-axis. This visual representation provides a powerful tool for understanding the polynomial's behavior and its relationship to its factors. The process of finding zeros is a fundamental skill in algebra and calculus, with applications in various fields, including engineering, physics, and economics. By mastering these techniques, we can effectively analyze and solve polynomial equations, gaining valuable insights into the behavior of mathematical functions. This comprehensive approach to finding zeros highlights the interconnectedness of algebraic and graphical concepts, fostering a deeper understanding of polynomial functions and their applications.
