Finding Zeros Of Polynomial Function F(x) = (x-7)(x+4)(3x-2)

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In mathematics, finding the zeros of a function is a fundamental task, especially when dealing with polynomial functions. Zeros, also known as roots or x-intercepts, are the values of x for which the function f(x) equals zero. Identifying these zeros is crucial for understanding the behavior of the function, graphing it accurately, and solving related equations. In this comprehensive guide, we will delve into the process of finding the zeros of the polynomial function f(x) = (x-7)(x+4)(3x-2). This function is presented in factored form, which simplifies the task of finding its zeros. We will explore the underlying principles, provide a step-by-step solution, and discuss the significance of zeros in the broader context of polynomial functions. This article aims to provide a clear and detailed explanation, suitable for students, educators, and anyone interested in enhancing their understanding of algebra and polynomial functions. By the end of this guide, you will have a solid grasp of how to identify zeros and appreciate their role in mathematical analysis.

Understanding Zeros of a Function

Zeros of a function, as mentioned earlier, are the values of the independent variable (typically x) that make the function's output equal to zero. Graphically, these are the points where the function's graph intersects the x-axis. For a polynomial function, the zeros provide valuable information about the function's behavior, such as its turning points and overall shape. The number of zeros a polynomial function has is related to its degree; a polynomial of degree n can have at most n zeros, counting multiplicities. Multiplicity refers to the number of times a particular zero appears as a root of the polynomial. For instance, if a factor (x - a) appears twice in the factored form of the polynomial, then a is a zero with a multiplicity of 2. This multiplicity affects how the graph of the function behaves at that zero; a zero with an even multiplicity will cause the graph to touch the x-axis and turn around, while a zero with an odd multiplicity will cause the graph to cross the x-axis. Understanding these concepts is crucial for analyzing and interpreting polynomial functions effectively. Zeros are not just mathematical curiosities; they have practical applications in various fields, including engineering, physics, and economics, where polynomial functions are used to model real-world phenomena. Therefore, mastering the techniques for finding zeros is an essential skill in mathematical problem-solving.

Factored Form and the Zero Product Property

The factored form of a polynomial is an expression of the polynomial as a product of its factors. For example, f(x) = (x-7)(x+4)(3x-2) is a polynomial in factored form. The factored form is particularly useful for identifying the zeros of the function because of the Zero Product Property. The Zero Product Property states that if the product of several factors is zero, then at least one of the factors must be zero. Mathematically, if A * B = 0, then either A = 0 or B = 0 (or both). This property is the cornerstone of finding zeros when a polynomial is in factored form. By setting each factor equal to zero and solving for x, we can easily determine the zeros of the polynomial. In our example, the factors are (x-7), (x+4), and (3x-2). Setting each of these equal to zero will give us the zeros of the function f(x). This method significantly simplifies the process compared to finding zeros of a polynomial in its expanded form, which often requires more complex techniques like synthetic division or the quadratic formula. The factored form directly reveals the zeros, making it an invaluable tool for polynomial analysis. Understanding and applying the Zero Product Property is essential for anyone working with polynomial functions, as it provides a straightforward way to find the roots of the equation and understand the behavior of the function.

Step-by-Step Solution for f(x) = (x-7)(x+4)(3x-2)

To find the zeros of the function f(x) = (x-7)(x+4)(3x-2), we will apply the Zero Product Property. This involves setting each factor equal to zero and solving for x. Let’s break down the process step by step:

  1. Identify the factors: The given function is already in factored form: f(x) = (x-7)(x+4)(3x-2). The factors are (x-7), (x+4), and (3x-2).
  2. Set each factor equal to zero:
    • x - 7 = 0
    • x + 4 = 0
    • 3x - 2 = 0
  3. Solve each equation for x:
    • For x - 7 = 0, add 7 to both sides to get x = 7.
    • For x + 4 = 0, subtract 4 from both sides to get x = -4.
    • For 3x - 2 = 0, add 2 to both sides to get 3x = 2, then divide by 3 to get x = 2/3.
  4. List the zeros: The zeros of the function f(x) are 7, -4, and 2/3. These are the x-values where the function's graph intersects the x-axis. Each of these zeros corresponds to a root of the polynomial equation f(x) = 0. The solutions represent the x-values that make the function equal to zero, providing key information about the function's behavior and graph. By following these steps, we have successfully identified the zeros of the given polynomial function. This method is highly efficient for polynomials presented in factored form and underscores the importance of the Zero Product Property in algebra.

Verifying the Zeros

After finding the zeros of a function, it's a good practice to verify them to ensure accuracy. This can be done by substituting each zero back into the original function and checking if the result is indeed zero. For f(x) = (x-7)(x+4)(3x-2), we found the zeros to be x = 7, x = -4, and x = 2/3. Let’s verify each:

  1. Verify x = 7:
    • f(7) = (7-7)(7+4)(3(7)-2) = (0)(11)(19) = 0
    • Since f(7) = 0, x = 7 is indeed a zero.
  2. Verify x = -4:
    • f(-4) = (-4-7)(-4+4)(3(-4)-2) = (-11)(0)(-14) = 0
    • Since f(-4) = 0, x = -4 is also a zero.
  3. Verify x = 2/3:
    • f(2/3) = (2/3 - 7)(2/3 + 4)(3(2/3) - 2) = (2/3 - 7)(2/3 + 4)(2 - 2) = (2/3 - 7)(2/3 + 4)(0) = 0
    • Since f(2/3) = 0, x = 2/3 is a zero as well.

By verifying each zero, we confirm that our solutions are correct. This process not only ensures accuracy but also reinforces the understanding of what it means for a value to be a zero of a function. Verification is a crucial step in mathematical problem-solving, providing confidence in the results and highlighting any potential errors in the calculations. In this case, the verification process confirms that 7, -4, and 2/3 are the zeros of the function f(x) = (x-7)(x+4)(3x-2).

Graphical Interpretation of Zeros

The graphical interpretation of zeros provides a visual understanding of what zeros represent. The zeros of a function are the points where the graph of the function intersects the x-axis. In other words, they are the x-coordinates of the points where the graph crosses or touches the horizontal axis. For the function f(x) = (x-7)(x+4)(3x-2), we found the zeros to be x = 7, x = -4, and x = 2/3. This means that the graph of f(x) will intersect the x-axis at these three points. When graphing the function, these zeros serve as crucial reference points. They help in sketching the curve and understanding the intervals where the function is positive or negative. The zeros divide the x-axis into intervals, and within each interval, the function's value will be either consistently positive or consistently negative. Furthermore, the behavior of the graph near the zeros is influenced by the multiplicity of the zeros. In this case, each zero has a multiplicity of 1 (since each factor appears only once), meaning the graph will cross the x-axis at each zero. Visualizing the zeros on a graph enhances the comprehension of the function's overall behavior and its relationship with the x-axis. The graphical representation complements the algebraic approach, providing a comprehensive understanding of the zeros of the function and their significance in the context of the function's graph.

Conclusion

In conclusion, identifying the zeros of a polynomial function is a crucial skill in algebra and calculus. For the function f(x) = (x-7)(x+4)(3x-2), we successfully found the zeros to be x = 7, x = -4, and x = 2/3 by applying the Zero Product Property. This involved setting each factor of the polynomial equal to zero and solving for x. We also emphasized the importance of verifying the zeros by substituting them back into the original function to ensure accuracy. The graphical interpretation of zeros further illustrated their significance as the x-intercepts of the function's graph. Understanding zeros is essential for analyzing the behavior of polynomial functions, sketching their graphs, and solving related equations. The steps outlined in this guide provide a clear and systematic approach for finding zeros, which can be applied to other polynomial functions as well. Mastering this skill will enhance your problem-solving abilities and deepen your understanding of mathematical functions. By recognizing the connection between algebraic solutions and graphical representations, you gain a more complete and intuitive grasp of the concepts. This comprehensive understanding is invaluable for further studies in mathematics and its applications in various fields.