Finding Zeros Of Polynomial P(x) = (2x² - 9x + 7)(x - 2)

In this article, we embark on a mathematical journey to find the zeros of a given polynomial function. Our focus is on the polynomial p(x) = (2x² - 9x + 7)(x - 2). The zeros of a polynomial, also known as roots or x-intercepts, are the values of x for which the polynomial evaluates to zero. These points hold significant importance in understanding the behavior and characteristics of the polynomial function. By determining these zeros, we can gain insights into the graph of the polynomial, its factors, and its overall properties. This exploration will involve factoring techniques, the quadratic formula, and graphical representation to provide a comprehensive understanding of how to identify and interpret the zeros of a polynomial.

H2: Understanding Polynomial Zeros

To effectively find the zeros of the polynomial, it's crucial to first grasp the fundamental concept of what polynomial zeros represent. In essence, the zeros of a polynomial p(x) are the values of x that make the polynomial equal to zero, i.e., p(x) = 0. These zeros are also known as the roots of the polynomial equation. Geometrically, the zeros correspond to the x-intercepts of the polynomial's graph, where the curve intersects the x-axis. Each zero represents a point where the polynomial changes its sign, transitioning from positive to negative or vice versa. Understanding the nature and multiplicity of zeros provides valuable information about the behavior of the polynomial function. For instance, a zero with multiplicity 2 indicates that the graph touches the x-axis at that point but doesn't cross it, while a zero with multiplicity 1 implies a crossing of the x-axis. The number of zeros a polynomial has is directly related to its degree, with a polynomial of degree n having at most n zeros (counting multiplicities). This connection between zeros and the degree of the polynomial is a cornerstone of polynomial theory, enabling us to predict the maximum number of roots a polynomial equation can possess. In the subsequent sections, we will apply this knowledge to the specific polynomial p(x) = (2x² - 9x + 7)(x - 2) to determine its zeros and their significance.

H2: Factoring the Polynomial

The first step in finding the zeros of the polynomial p(x) = (2x² - 9x + 7)(x - 2) is to factor it completely. Factoring a polynomial involves expressing it as a product of simpler polynomials, typically linear or quadratic factors. In this case, the polynomial is already partially factored, with one factor being (x - 2). Our primary task is to factor the quadratic expression (2x² - 9x + 7). There are several techniques for factoring quadratic expressions, including trial and error, grouping, and using the quadratic formula. For this particular quadratic, we can employ the factoring by grouping method. We look for two numbers that multiply to (2 * 7 = 14) and add up to -9. These numbers are -2 and -7. We can then rewrite the middle term, -9x, as -2x - 7x. This allows us to split the quadratic into four terms: 2x² - 2x - 7x + 7. Next, we group the first two terms and the last two terms: (2x² - 2x) + (-7x + 7). We factor out the greatest common factor (GCF) from each group. From the first group, we can factor out 2x, and from the second group, we can factor out -7. This yields 2x(x - 1) - 7(x - 1). Now, we observe that (x - 1) is a common factor in both terms. Factoring out (x - 1) gives us (x - 1)(2x - 7). Therefore, the factored form of the quadratic expression (2x² - 9x + 7) is (x - 1)(2x - 7). Combining this with the existing factor (x - 2), the completely factored form of the polynomial p(x) is (x - 1)(2x - 7)(x - 2). This factored form is crucial because it directly reveals the zeros of the polynomial, as we will explore in the next section.

H2: Determining the Zeros

With the polynomial p(x) = (2x² - 9x + 7)(x - 2) factored into (x - 1)(2x - 7)(x - 2), finding the zeros becomes a straightforward process. The zeros of a polynomial are the values of x that make the polynomial equal to zero. In factored form, this occurs when any of the factors equal zero. Therefore, to find the zeros, we set each factor equal to zero and solve for x. We have three factors: (x - 1), (2x - 7), and (x - 2). Setting the first factor (x - 1) equal to zero gives us x - 1 = 0, which implies x = 1. This means that x = 1 is one of the zeros of the polynomial. Next, we set the second factor (2x - 7) equal to zero: 2x - 7 = 0. Adding 7 to both sides gives 2x = 7. Dividing both sides by 2 yields x = 7/2 or x = 3.5. Thus, x = 3.5 is another zero of the polynomial. Finally, setting the third factor (x - 2) equal to zero gives us x - 2 = 0, which implies x = 2. Therefore, x = 2 is the third zero of the polynomial. In summary, the zeros of the polynomial p(x) = (2x² - 9x + 7)(x - 2) are x = 1, x = 3.5, and x = 2. These zeros represent the points where the graph of the polynomial intersects the x-axis. Understanding these zeros is essential for sketching the graph of the polynomial and analyzing its behavior. In the subsequent section, we will explore the graphical representation of the polynomial and how these zeros are visually depicted.

H2: Plotting the Zeros and Graphing the Polynomial

Once we have determined the zeros of the polynomial p(x) = (2x² - 9x + 7)(x - 2), which are x = 1, x = 2, and x = 3.5, we can proceed to plot these zeros on a graph. These zeros represent the x-intercepts of the polynomial, which are the points where the graph of the polynomial intersects the x-axis. To plot these zeros, we locate the corresponding points on the x-axis and mark them. The point (1, 0) corresponds to the zero x = 1, the point (2, 0) corresponds to the zero x = 2, and the point (3.5, 0) corresponds to the zero x = 3.5. These three points are the x-intercepts of the polynomial graph. To sketch the graph of the polynomial, we also need to consider the degree and the leading coefficient of the polynomial. The polynomial p(x) is a cubic polynomial (degree 3) because when we expand the factored form (x - 1)(2x - 7)(x - 2), the highest power of x will be 3. The leading coefficient is the coefficient of the term with the highest power of x. In this case, when we expand the factored form, the term with x³ will be 2x³, so the leading coefficient is 2, which is positive. For a cubic polynomial with a positive leading coefficient, the graph will start from the bottom left, pass through the zeros, and go towards the top right. Knowing the zeros and the general shape of the cubic polynomial, we can sketch the graph. The graph will cross the x-axis at x = 1, x = 2, and x = 3.5. Since all the zeros have multiplicity 1 (they appear only once in the factored form), the graph will cross the x-axis at each of these points. The graph will have a local maximum and a local minimum between the zeros. A more precise graph can be obtained by plotting additional points or using graphing software. However, by plotting the zeros and understanding the end behavior of the cubic polynomial, we can create a reasonable sketch of the graph. This graphical representation provides a visual confirmation of the zeros we calculated and helps in understanding the overall behavior of the polynomial function.

H2: Conclusion

In conclusion, finding the zeros of a polynomial is a fundamental concept in algebra with significant applications in various fields. In this article, we successfully determined the zeros of the polynomial p(x) = (2x² - 9x + 7)(x - 2) through factoring and solving for the roots. We identified the zeros as x = 1, x = 2, and x = 3.5. These zeros represent the x-intercepts of the polynomial's graph, providing key information about its behavior. We also discussed how to plot these zeros on a graph and sketch the polynomial function, considering its degree and leading coefficient. The process of finding zeros involves factoring the polynomial, setting each factor equal to zero, and solving for x. The factored form of the polynomial directly reveals its zeros, making this technique highly efficient. Graphical representation provides a visual confirmation of the zeros and helps in understanding the overall shape and behavior of the polynomial function. Understanding the zeros of a polynomial is crucial for solving polynomial equations, analyzing graphs, and solving real-world problems modeled by polynomials. This exploration of the polynomial p(x) = (2x² - 9x + 7)(x - 2) has provided a comprehensive understanding of how to find, interpret, and graphically represent the zeros of a polynomial function. The techniques and concepts discussed here can be applied to a wide range of polynomial functions, making this a valuable skill for anyone studying mathematics and related fields.