Finding The Zeros Of F(x) = X³ + 8x² + 5x - 50 A Step By Step Guide
In mathematics, finding the zeros of a polynomial function is a fundamental concept with widespread applications. The zeros, also known as roots or x-intercepts, are the values of x for which the function f(x) equals zero. These points hold significant importance in understanding the behavior and characteristics of the function's graph. When dealing with a cubic function like f(x) = x³ + 8x² + 5x - 50, identifying its zeros provides crucial insights into where the graph intersects the x-axis, its overall shape, and potential real-world applications. This article delves into a step-by-step approach to determine the zeros of the given cubic function, offering a comprehensive understanding of the methods and reasoning involved.
The Significance of Zeros
Zeros of a function are the points where the graph intersects the x-axis. These points are crucial because they indicate where the function's value is zero, providing essential information about the function's behavior and solutions to equations. The zeros can reveal where a process or quantity reaches a critical point, such as in optimization problems or when modeling physical phenomena. Understanding the zeros helps in sketching the graph of the function and in analyzing its properties, such as intervals of increase or decrease. This knowledge is also invaluable in various real-world applications, from engineering design to economic forecasting.
Methods for Finding Zeros
Finding the zeros of a polynomial function involves various techniques, each suited to different types of functions. For linear and quadratic functions, simple algebraic methods like factoring, the quadratic formula, or completing the square are often sufficient. However, cubic and higher-degree polynomials may require more advanced techniques such as the Rational Root Theorem, synthetic division, or numerical methods. The choice of method depends on the specific characteristics of the polynomial, including its degree and the nature of its coefficients. Understanding these methods is crucial for solving equations and gaining deeper insights into the behavior of polynomial functions.
1. The Rational Root Theorem: Identifying Potential Zeros
To find the zeros of the cubic function f(x) = x³ + 8x² + 5x - 50, we begin with the Rational Root Theorem. This theorem provides a systematic way to identify potential rational roots (zeros) of a polynomial. It states that if a polynomial has integer coefficients, then any rational root must be of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. In our case, the constant term is -50, and the leading coefficient is 1. By applying the Rational Root Theorem, we narrow down our search for zeros to a manageable set of possible values.
Applying the Rational Root Theorem
To apply the Rational Root Theorem effectively, we first need to list all the factors of the constant term (-50) and the leading coefficient (1). The factors of -50 are ±1, ±2, ±5, ±10, ±25, and ±50, while the factors of 1 are simply ±1. According to the theorem, any rational root of the polynomial must be a ratio of these factors. This means we need to consider all possible fractions formed by dividing the factors of -50 by the factors of 1. This process gives us a set of potential rational roots that we can test to see if they are actual zeros of the function.
Listing Potential Rational Roots
Based on the factors identified, the potential rational roots are: ±1, ±2, ±5, ±10, ±25, and ±50. These values are the candidates for the zeros of the cubic function f(x) = x³ + 8x² + 5x - 50. By listing these potential roots, we have a structured set of numbers to test, which significantly simplifies the process of finding the actual zeros. The next step involves testing these values using methods such as synthetic division or direct substitution to determine which ones make the function equal to zero.
2. Testing Potential Zeros: Synthetic Division
After identifying the potential rational roots using the Rational Root Theorem, the next step is to test these candidates to determine which ones are actual zeros of the function. Synthetic division is an efficient method for this purpose. Synthetic division is a streamlined process for dividing a polynomial by a linear factor (x - c), where c is a potential root. If the remainder of the synthetic division is zero, then c is a zero of the polynomial. This method not only confirms whether a number is a zero but also provides the quotient, which is a polynomial of lower degree that can be further analyzed.
The Process of Synthetic Division
To perform synthetic division, we set up a division table with the coefficients of the polynomial and the potential root we are testing. We then follow a series of steps involving bringing down the first coefficient, multiplying it by the potential root, adding the result to the next coefficient, and repeating this process until we reach the last coefficient. The final number in the bottom row is the remainder. If the remainder is zero, it confirms that the potential root is indeed a zero of the polynomial, and the other numbers in the bottom row represent the coefficients of the quotient polynomial. This quotient can then be used to find additional zeros or to factor the original polynomial further.
Applying Synthetic Division to f(x) = x³ + 8x² + 5x - 50
Let's apply synthetic division to our function f(x) = x³ + 8x² + 5x - 50. We'll start by testing the potential root -5. Setting up the synthetic division, we write the coefficients of the polynomial (1, 8, 5, -50) and the potential root (-5) in the appropriate positions. Performing the synthetic division, we find that the remainder is 0, which confirms that -5 is a zero of the function. The quotient polynomial obtained from this division is x² + 3x - 10. This result is crucial because it reduces our cubic function to a quadratic, which is much easier to solve. The quotient polynomial will help us find the remaining zeros of the original cubic function.
3. Factoring the Quotient: Finding Remaining Zeros
Once we have identified one zero of the cubic function, we can use the quotient obtained from synthetic division to find the remaining zeros. In our case, synthetic division with -5 yielded the quotient polynomial x² + 3x - 10. This is a quadratic polynomial, which can be factored using standard techniques. Factoring the quadratic is a critical step because it allows us to express the polynomial as a product of linear factors, each of which corresponds to a zero of the function. By setting each factor equal to zero, we can easily solve for the remaining zeros.
Factoring Techniques
There are several methods for factoring a quadratic polynomial, including trial and error, the quadratic formula, and completing the square. However, for the quadratic x² + 3x - 10, factoring by trial and error is the most straightforward approach. We look for two numbers that multiply to -10 and add to 3. These numbers are 5 and -2. Therefore, we can factor the quadratic as (x + 5)(x - 2). This factorization is crucial because it directly reveals the remaining zeros of the polynomial. Each factor represents a root of the equation, allowing us to quickly identify the x-values where the function equals zero.
Solving for the Remaining Zeros
Having factored the quadratic quotient as (x + 5)(x - 2), we now set each factor equal to zero to solve for x. This gives us the equations x + 5 = 0 and x - 2 = 0. Solving these equations, we find x = -5 and x = 2. These are the additional zeros of the original cubic function. However, we already knew that -5 was a zero from our synthetic division, so this confirms our previous finding and introduces a new zero at x = 2. This process demonstrates how factoring the quotient polynomial helps us uncover all the zeros of the higher-degree polynomial efficiently.
4. Identifying All Zeros: The Complete Solution
Now that we have factored the quotient polynomial and solved for its zeros, we can compile all the zeros of the original cubic function f(x) = x³ + 8x² + 5x - 50. We initially found one zero by using synthetic division with a potential rational root, and then we used the resulting quotient to find the remaining zeros. Combining these results gives us the complete set of zeros for the function, which is essential for understanding its behavior and graph. Identifying all zeros is the final step in solving the problem and provides a comprehensive solution.
Listing All Zeros
From the synthetic division, we identified x = -5 as a zero. Factoring the quotient polynomial x² + 3x - 10, we found the zeros x = -5 and x = 2. Therefore, the zeros of the cubic function f(x) = x³ + 8x² + 5x - 50 are -5 (with multiplicity 2) and 2. This complete set of zeros tells us where the graph of the function intersects the x-axis and provides critical information for sketching the graph and understanding the function's properties.
Representing Zeros as Coordinates
To represent these zeros as coordinates, we write them as points on the x-axis where the function's value is zero. Since the zeros are the x-values where f(x) = 0, the coordinates are of the form (x, 0). Therefore, the zeros -5 and 2 correspond to the coordinates (-5, 0) and (2, 0). These points are where the graph of the function intersects the x-axis, providing a visual representation of the zeros. Understanding how to represent zeros as coordinates is essential for graphing functions and interpreting their behavior in the coordinate plane.
5. Selecting the Correct Option: The Final Answer
After identifying all the zeros and representing them as coordinates, the final step is to select the correct option from the given choices. This involves comparing the coordinates we found with the options provided and choosing the one that accurately lists all the zeros. This step ensures that we have correctly applied the methods and arrived at the right solution. Selecting the correct option is the culmination of the problem-solving process and confirms our understanding of the concepts involved.
Reviewing the Options
Given the options:
A. (-5,0),(2,0)
B. (-5,0),(-2,0)
C. (-5,0),(-2,0),(5,0)
D. (-5,0),(2,0),(5,0)
We found the zeros to be -5 and 2, which correspond to the coordinates (-5, 0) and (2, 0). Comparing these coordinates with the given options, we can see that option A matches our solution. This confirms that we have correctly identified all the zeros of the function and represented them in the appropriate coordinate form. Option A is therefore the correct answer.
Conclusion: Option A is the Correct Answer
Based on our step-by-step solution, the coordinates that best represent all zeros of the graph of f(x) = x³ + 8x² + 5x - 50 are (-5, 0) and (2, 0). Therefore, the correct answer is A. (-5, 0), (2, 0). This solution demonstrates the application of the Rational Root Theorem, synthetic division, and factoring techniques to find the zeros of a cubic function. Understanding these methods is crucial for solving polynomial equations and analyzing the behavior of polynomial functions in various mathematical and real-world contexts.
In conclusion, finding the zeros of a function, particularly a cubic function like f(x) = x³ + 8x² + 5x - 50, is a fundamental skill in mathematics. This comprehensive guide has walked through a detailed process, starting with the Rational Root Theorem to identify potential zeros, followed by synthetic division to confirm these zeros and reduce the polynomial's degree, and then factoring the quotient to find the remaining zeros. Representing these zeros as coordinates provides a clear understanding of where the function intersects the x-axis, a crucial aspect of graphing and analyzing functions. The correct answer to the question, A. (-5, 0), (2, 0), highlights the importance of systematic problem-solving techniques in mathematics.
Recap of Key Concepts
The Importance of Zeros
Zeros of a function are the points where the graph intersects the x-axis. They provide critical information about the function's behavior and solutions to equations. Understanding the zeros helps in sketching the graph of the function and in analyzing its properties, such as intervals of increase or decrease. This knowledge is also invaluable in various real-world applications, from engineering design to economic forecasting.
The Rational Root Theorem
The Rational Root Theorem provides a systematic way to identify potential rational roots of a polynomial. It states that if a polynomial has integer coefficients, then any rational root must be of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. This theorem helps narrow down the search for zeros to a manageable set of possible values.
Synthetic Division
Synthetic division is an efficient method for dividing a polynomial by a linear factor (x - c), where c is a potential root. If the remainder of the synthetic division is zero, then c is a zero of the polynomial. This method not only confirms whether a number is a zero but also provides the quotient, which is a polynomial of lower degree that can be further analyzed.
Factoring Techniques
Factoring is a crucial step in finding the zeros of a polynomial. For quadratic polynomials, techniques such as trial and error, the quadratic formula, and completing the square can be used. Factoring a polynomial allows it to be expressed as a product of linear factors, each of which corresponds to a zero of the function. By setting each factor equal to zero, we can easily solve for the remaining zeros.
Final Thoughts
Mastering the art of finding zeros is essential for success in algebra and beyond. The techniques discussed in this article, including the Rational Root Theorem, synthetic division, and factoring, are powerful tools for solving polynomial equations and understanding the behavior of polynomial functions. By practicing these methods and understanding the underlying concepts, students can confidently tackle complex problems and gain a deeper appreciation for the beauty and power of mathematics. Whether in academic pursuits or real-world applications, the ability to find zeros is a valuable asset that opens doors to further exploration and discovery.
