Finding Zeros Of Polynomial P(x)=(2x^2-9x+7)(x-2) A Step-by-Step Guide

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To find the zeros of the polynomial p(x) = (2x² - 9x + 7)(x - 2), we embark on a journey through algebraic techniques, factoring, and the Zero Product Property. This exploration not only reveals the solutions but also deepens our understanding of polynomial behavior and its graphical representation. In this detailed guide, we will walk through the step-by-step process of finding the zeros of the given polynomial, providing explanations and insights along the way. Understanding how to find the zeros of a polynomial is a fundamental skill in algebra and calculus, with applications spanning various fields like engineering, physics, and computer science. The zeros, also known as roots or x-intercepts, are the values of x for which the polynomial p(x) equals zero. These points are crucial for analyzing the function's behavior, determining where it crosses the x-axis, and solving related equations. The process of finding the zeros often involves factoring the polynomial, applying the quadratic formula, or using numerical methods for more complex cases. In our example, we'll focus on factoring and the Zero Product Property, which are powerful tools for polynomials that can be expressed as products of simpler expressions. This method allows us to break down the problem into smaller, more manageable parts, making it easier to identify the values of x that make the entire polynomial equal to zero. Let's dive into the specifics of our polynomial and uncover its zeros through a systematic and clear approach.

Factoring the Quadratic Expression

The cornerstone of finding the zeros of p(x) lies in factoring the quadratic expression 2x² - 9x + 7. Factoring is the process of breaking down a polynomial into a product of simpler polynomials. In this case, we aim to express 2x² - 9x + 7 as a product of two binomials. This step is crucial because it allows us to apply the Zero Product Property later on. The Zero Product Property states that if the product of two or more factors is zero, then at least one of the factors must be zero. This principle is the backbone of solving polynomial equations once they are factored. There are several techniques for factoring quadratic expressions, including trial and error, grouping, and using the quadratic formula to find the roots and then constructing the factors. For 2x² - 9x + 7, we can use the grouping method, which involves finding two numbers that multiply to the product of the leading coefficient (2) and the constant term (7), which is 14, and add up to the middle coefficient (-9). These two numbers are -2 and -7. We then rewrite the middle term, -9x, as the sum of -2x and -7x, allowing us to factor by grouping. This method transforms the quadratic expression into a form where common factors can be easily identified and extracted. The factored form provides direct insight into the roots of the quadratic, as each factor corresponds to a potential zero of the polynomial. Once we have factored 2x² - 9x + 7, we can combine it with the other factor in p(x), which is (x - 2), to set the stage for finding all the zeros of the polynomial. This process highlights the power of factoring in simplifying complex expressions and revealing their underlying structure, making it a fundamental technique in algebra.

Step-by-Step Factoring Process

Let's delve into the step-by-step process of factoring the quadratic expression 2x² - 9x + 7. This process involves transforming the quadratic into a product of two binomials, which is a crucial step towards finding the zeros of the polynomial. First, we identify the coefficients: the leading coefficient is 2, the middle coefficient is -9, and the constant term is 7. Our goal is to find two numbers that multiply to the product of the leading coefficient and the constant term (2 * 7 = 14) and add up to the middle coefficient (-9). These two numbers are -2 and -7. Next, we rewrite the middle term, -9x, as the sum of -2x and -7x. This gives us the expression 2x² - 2x - 7x + 7. Now, we factor by grouping. We group the first two terms and the last two terms: (2x² - 2x) + (-7x + 7). From the first group, we can factor out 2x, leaving us with 2x(x - 1). From the second group, we can factor out -7, leaving us with -7(x - 1). Notice that both groups now have a common factor of (x - 1). We factor out (x - 1) from the entire expression, which gives us (x - 1)(2x - 7). Thus, the factored form of 2x² - 9x + 7 is (x - 1)(2x - 7). This factored form is essential because it allows us to apply the Zero Product Property. By setting each factor equal to zero, we can find the values of x that make the quadratic expression equal to zero. This process not only simplifies the task of finding the zeros but also provides a clear understanding of the roots of the polynomial. Factoring is a fundamental skill in algebra, and mastering this technique is crucial for solving polynomial equations and analyzing their behavior. The step-by-step approach ensures clarity and accuracy, making it easier to tackle more complex factoring problems in the future.

Applying the Zero Product Property

With the polynomial p(x) = (2x² - 9x + 7)(x - 2) factored into p(x) = (x - 1)(2x - 7)(x - 2), we can now apply the Zero Product Property. This property is the key to unlocking the zeros of the polynomial. The Zero Product Property states that if the product of several factors is zero, then at least one of the factors must be zero. In our case, the factors are (x - 1), (2x - 7), and (x - 2). By setting each of these factors equal to zero, we create simple linear equations that can be easily solved for x. This step is crucial because it transforms the problem of finding the zeros of a cubic polynomial into a series of simpler, more manageable equations. Each solution we find corresponds to a zero of the polynomial, which is a value of x that makes p(x) equal to zero. These zeros are also the x-intercepts of the polynomial's graph, providing valuable information about the function's behavior. The Zero Product Property is a powerful tool in algebra, allowing us to solve polynomial equations by breaking them down into smaller parts. It highlights the importance of factoring, as it is the factored form of the polynomial that allows us to apply this property effectively. By systematically setting each factor to zero and solving for x, we can identify all the zeros of the polynomial, gaining a complete understanding of its roots and its behavior on the coordinate plane. This process not only solves the problem at hand but also reinforces the fundamental principles of algebra and equation solving.

Solving for x

To solve for x, we set each factor equal to zero and solve the resulting linear equations. This is a straightforward process that yields the zeros of the polynomial. First, we set (x - 1) = 0. Adding 1 to both sides, we find x = 1. This is one of the zeros of the polynomial. Next, we set (2x - 7) = 0. Adding 7 to both sides gives us 2x = 7. Dividing both sides by 2, we find x = 7/2 or x = 3.5. This is another zero of the polynomial. Finally, we set (x - 2) = 0. Adding 2 to both sides, we find x = 2. This is the third and final zero of the polynomial. Thus, the zeros of the polynomial p(x) = (x - 1)(2x - 7)(x - 2) are x = 1, x = 3.5, and x = 2. These values are the points where the graph of the polynomial intersects the x-axis. Understanding how to solve for x in this context is crucial for analyzing polynomial functions and their behavior. Each zero corresponds to a root of the polynomial equation, and together they provide a complete picture of where the function equals zero. This process reinforces the importance of algebraic manipulation and equation solving skills. By systematically setting each factor to zero and solving for x, we can confidently identify all the zeros of the polynomial, gaining a thorough understanding of its properties and behavior.

Verifying the Zeros

After finding the zeros of the polynomial, it's essential to verify them to ensure accuracy. Verification involves substituting each zero back into the original polynomial p(x) = (2x² - 9x + 7)(x - 2) and confirming that the result is indeed zero. This step is crucial for catching any potential errors in the factoring or solving process. It also reinforces our understanding of what it means for a value to be a zero of a polynomial. A zero, by definition, is a value of x that makes the polynomial equal to zero. By substituting our calculated zeros back into the polynomial, we are directly testing this definition. This process not only confirms our solutions but also deepens our understanding of the relationship between the zeros of a polynomial and its factored form. If the polynomial evaluates to zero for each substituted zero, we can be confident in our results. If not, it indicates a potential error that needs to be revisited. Verification is a fundamental step in mathematical problem-solving, ensuring that our answers are correct and reliable. It also provides an opportunity to review our work and reinforce the concepts involved. By verifying the zeros of our polynomial, we not only confirm our solutions but also strengthen our overall understanding of polynomial functions and their properties. This step is a testament to the importance of accuracy and attention to detail in mathematics.

Substitution and Confirmation

Let's verify the zeros we found: x = 1, x = 3.5, and x = 2. We will substitute each value into the original polynomial p(x) = (2x² - 9x + 7)(x - 2) and check if the result is zero. First, let's substitute x = 1: p(1) = (2(1)² - 9(1) + 7)(1 - 2) = (2 - 9 + 7)(-1) = (0)(-1) = 0. This confirms that x = 1 is a zero of the polynomial. Next, let's substitute x = 3.5 or x = 7/2: p(3.5) = (2(3.5)² - 9(3.5) + 7)(3.5 - 2) = (2(12.25) - 31.5 + 7)(1.5) = (24.5 - 31.5 + 7)(1.5) = (0)(1.5) = 0. This confirms that x = 3.5 is also a zero of the polynomial. Finally, let's substitute x = 2: p(2) = (2(2)² - 9(2) + 7)(2 - 2) = (2(4) - 18 + 7)(0) = (8 - 18 + 7)(0) = (-3)(0) = 0. This confirms that x = 2 is a zero of the polynomial as well. Since the polynomial evaluates to zero for each of the substituted values, we can confidently conclude that x = 1, x = 3.5, and x = 2 are indeed the zeros of the polynomial p(x) = (2x² - 9x + 7)(x - 2). This verification process underscores the importance of double-checking our work and ensuring the accuracy of our solutions. It also reinforces the fundamental concept of a zero of a polynomial as a value that makes the polynomial equal to zero. By systematically substituting and confirming, we not only validate our answers but also strengthen our understanding of polynomial functions and their properties.

Conclusion

In conclusion, by meticulously factoring the polynomial p(x) = (2x² - 9x + 7)(x - 2), applying the Zero Product Property, and verifying our solutions, we have successfully identified the zeros of the polynomial as x = 1, x = 3.5, and x = 2. This process demonstrates the power of algebraic techniques in solving polynomial equations and understanding their behavior. Factoring allowed us to break down the complex polynomial into simpler factors, making it easier to find the values of x that make the polynomial equal to zero. The Zero Product Property then provided the crucial link between the factored form and the zeros, enabling us to systematically solve for x. Finally, verification ensured the accuracy of our solutions, reinforcing the importance of carefulness and attention to detail in mathematics. This comprehensive approach not only solves the specific problem but also provides a framework for tackling similar problems in the future. Understanding how to find the zeros of a polynomial is a fundamental skill in algebra, with applications in various fields. The zeros, or roots, of a polynomial provide valuable information about its graph, its behavior, and its relationship to other mathematical concepts. By mastering the techniques of factoring, applying the Zero Product Property, and verifying solutions, we can confidently analyze and solve polynomial equations, gaining a deeper appreciation for the elegance and power of mathematics. This journey through polynomial zeros has highlighted the interconnectedness of different algebraic concepts and the importance of a systematic approach to problem-solving.