Sign Analysis Of Polynomial Function F(x) In The Interval (-7/3, -3/5)

In the realm of mathematics, understanding the behavior of polynomial functions is crucial. These functions, defined by algebraic expressions involving variables raised to non-negative integer powers, exhibit a rich array of properties and patterns. Among these properties, the sign of a polynomial function over a specific interval holds significant importance. This analysis delves into the intricacies of determining the sign of a polynomial function, using the specific example of f(x) = (5x+3)(x-2)(3x+7)(x+5). This function, expressed as a product of linear factors, presents a unique opportunity to explore the relationship between its zeros and its sign across different intervals. The zeros of a polynomial function, the points where the function intersects the x-axis, play a pivotal role in determining its sign. These zeros act as boundaries, dividing the number line into intervals where the function maintains a consistent sign – either positive or negative. By identifying these zeros and analyzing the behavior of the function within each interval, we can gain a comprehensive understanding of its sign pattern. This process involves a careful examination of the factors that constitute the polynomial function. Each linear factor, of the form (ax + b), contributes a zero at x = -b/a. The sign of this factor changes at its zero, transitioning from negative to positive or vice versa, depending on the sign of the coefficient 'a'. By considering the combined effect of all the factors, we can determine the overall sign of the polynomial function in each interval. This analysis not only provides insights into the behavior of the function but also lays the groundwork for solving inequalities and understanding the function's graph. Furthermore, it highlights the fundamental connection between algebra and calculus, as the sign of a function is closely related to its derivative and its increasing or decreasing nature. In this article, we will embark on a detailed exploration of the sign of the polynomial function f(x) = (5x+3)(x-2)(3x+7)(x+5), focusing on the interval (-7/3, -3/5). We will meticulously analyze the factors, identify the zeros, and construct a sign chart to determine the function's sign within this specific interval. This step-by-step approach will not only provide a solution to the given problem but also equip readers with a comprehensive understanding of the techniques involved in analyzing the sign of polynomial functions. This knowledge is invaluable for students, educators, and anyone interested in delving deeper into the world of mathematics.

Identifying the Zeros and Intervals of f(x)

To effectively determine the sign of the polynomial function f(x) = (5x+3)(x-2)(3x+7)(x+5), a crucial first step is to pinpoint its zeros. The zeros, also known as roots, are the values of x for which the function f(x) equals zero. These points are particularly significant because they mark the locations where the function's graph intersects the x-axis, and they serve as boundaries that divide the number line into intervals where the function maintains a consistent sign – either positive or negative. As the function is expressed as a product of linear factors, the zeros can be easily identified by setting each factor equal to zero and solving for x. This process stems from the fundamental principle that a product is zero if and only if at least one of its factors is zero. Therefore, to find the zeros of f(x), we set each of the factors (5x+3), (x-2), (3x+7), and (x+5) equal to zero. Solving 5x+3 = 0 yields x = -3/5. Similarly, solving x-2 = 0 gives x = 2; solving 3x+7 = 0 results in x = -7/3; and solving x+5 = 0 gives x = -5. Thus, the zeros of f(x) are x = -5, -7/3, -3/5, and 2. These zeros, arranged in ascending order, are crucial for constructing a sign chart, which is a visual tool that helps us analyze the sign of the function across different intervals. The sign chart consists of a number line with the zeros marked on it. These zeros divide the number line into distinct intervals. In this case, the zeros -5, -7/3, -3/5, and 2 divide the number line into five intervals: (-∞, -5), (-5, -7/3), (-7/3, -3/5), (-3/5, 2), and (2, ∞). Within each of these intervals, the function f(x) maintains a consistent sign, either positive or negative. This is because the sign of the function can only change at its zeros. To determine the sign of f(x) in each interval, we can select a test value within that interval and evaluate the function at that value. The sign of f(x) at the test value will be the sign of the function throughout the entire interval. Alternatively, we can analyze the sign of each linear factor within each interval and then multiply the signs together to determine the sign of f(x). This approach leverages the fact that the sign of a product is determined by the signs of its factors. The intervals we have identified are the key to understanding the sign behavior of the polynomial function. By analyzing the sign of f(x) in each interval, we can gain valuable insights into its graph and its overall behavior. This analysis is particularly useful for solving inequalities involving polynomial functions, as it allows us to identify the intervals where the function is positive or negative. Moreover, it forms a fundamental concept in calculus, where the sign of a function's derivative is used to determine the function's increasing or decreasing behavior.

Constructing the Sign Chart for f(x)

Constructing a sign chart is a pivotal step in analyzing the sign of the polynomial function f(x) = (5x+3)(x-2)(3x+7)(x+5). This visual tool provides a systematic way to determine the sign of the function across different intervals defined by its zeros. The sign chart is essentially a number line with the zeros of the function marked on it. These zeros, as we previously identified, are x = -5, -7/3, -3/5, and 2. These values are placed on the number line in ascending order, dividing it into five distinct intervals: (-∞, -5), (-5, -7/3), (-7/3, -3/5), (-3/5, 2), and (2, ∞). To construct the sign chart, we need to analyze the sign of each linear factor of f(x) within each interval. The linear factors are (5x+3), (x-2), (3x+7), and (x+5). The sign of each factor changes at its corresponding zero. For instance, the factor (5x+3) is negative when x < -3/5 and positive when x > -3/5. Similarly, (x-2) is negative when x < 2 and positive when x > 2; (3x+7) is negative when x < -7/3 and positive when x > -7/3; and (x+5) is negative when x < -5 and positive when x > -5. We represent these sign changes on the sign chart by indicating the sign of each factor in each interval. Typically, a '+' sign is used to indicate a positive value, and a '-' sign is used to indicate a negative value. For example, in the interval (-∞, -5), all four factors are negative. Therefore, we would mark '-' signs for each factor in this interval. In the interval (-5, -7/3), the factor (x+5) is positive, while the other three factors remain negative. In the interval (-7/3, -3/5), the factors (x+5) and (3x+7) are positive, while (5x+3) and (x-2) are negative. In the interval (-3/5, 2), the factors (x+5), (3x+7), and (5x+3) are positive, while (x-2) is negative. Finally, in the interval (2, ∞), all four factors are positive. Once we have determined the sign of each factor in each interval, we can determine the sign of f(x) by multiplying the signs of the factors together. The sign of a product is positive if there is an even number of negative factors and negative if there is an odd number of negative factors. For example, in the interval (-∞, -5), there are four negative factors, so the sign of f(x) is positive. In the interval (-5, -7/3), there are three negative factors, so the sign of f(x) is negative. In the interval (-7/3, -3/5), there are two negative factors, so the sign of f(x) is positive. In the interval (-3/5, 2), there is one negative factor, so the sign of f(x) is negative. Finally, in the interval (2, ∞), there are no negative factors, so the sign of f(x) is positive. The sign chart, now complete, provides a clear and concise visual representation of the sign of f(x) across all intervals. This information is invaluable for solving inequalities, sketching the graph of f(x), and understanding its overall behavior.

Determining the Sign of f(x) in the Interval (-7/3, -3/5)

Having established the sign chart for the polynomial function f(x) = (5x+3)(x-2)(3x+7)(x+5), we are now equipped to determine the sign of f(x) specifically within the interval (-7/3, -3/5). This interval lies between two of the function's zeros, namely x = -7/3 and x = -3/5. The sign chart provides a comprehensive overview of the sign of f(x) across all intervals, but we will focus our attention on this particular interval to address the question at hand. The key to determining the sign of f(x) in this interval lies in analyzing the sign of each of its linear factors: (5x+3), (x-2), (3x+7), and (x+5). We need to ascertain whether each factor is positive or negative within the interval (-7/3, -3/5). Let's consider each factor individually. The factor (5x+3) is zero when x = -3/5. Since we are considering the interval (-7/3, -3/5), which consists of values of x less than -3/5, the factor (5x+3) will be negative within this interval. This is because for any x less than -3/5, the expression 5x+3 will result in a negative value. Next, let's examine the factor (x-2). This factor is zero when x = 2. Since all values of x in the interval (-7/3, -3/5) are significantly less than 2, the factor (x-2) will be negative within this interval. For any x less than 2, subtracting 2 will result in a negative value. Now, let's consider the factor (3x+7). This factor is zero when x = -7/3. Within the interval (-7/3, -3/5), x is greater than -7/3. Therefore, the factor (3x+7) will be positive within this interval. For any x greater than -7/3, the expression 3x+7 will result in a positive value. Finally, let's analyze the factor (x+5). This factor is zero when x = -5. Since all values of x in the interval (-7/3, -3/5) are greater than -5, the factor (x+5) will be positive within this interval. For any x greater than -5, adding 5 will result in a positive value. Now that we have determined the sign of each factor within the interval (-7/3, -3/5), we can determine the sign of f(x) by multiplying the signs together. We have two negative factors (5x+3) and (x-2), and two positive factors (3x+7) and (x+5). The product of two negative numbers is positive, and the product of two positive numbers is positive. Therefore, the product of all four factors, which is f(x), will be positive within the interval (-7/3, -3/5). In conclusion, by analyzing the signs of the linear factors of f(x) within the interval (-7/3, -3/5), we have determined that the sign of f(x) is positive in this interval. This result aligns with the information provided by the sign chart, which visually represents the sign of f(x) across all intervals defined by its zeros. This analysis demonstrates the power of sign charts in understanding the behavior of polynomial functions and their signs across different intervals.

Conclusion: Sign of f(x) in the Interval (-7/3, -3/5)

In this comprehensive exploration, we have delved into the intricacies of analyzing the sign of the polynomial function f(x) = (5x+3)(x-2)(3x+7)(x+5), with a particular focus on the interval (-7/3, -3/5). Through a systematic approach, we have successfully determined the sign of f(x) within this interval. Our journey began with the crucial step of identifying the zeros of f(x). These zeros, x = -5, -7/3, -3/5, and 2, serve as the boundary points that divide the number line into intervals where the function maintains a consistent sign. Understanding the zeros is fundamental to constructing a sign chart, a powerful tool for visualizing the sign behavior of a function. The sign chart, which we meticulously constructed, allowed us to track the sign of each linear factor of f(x) across different intervals. By analyzing the sign of each factor – (5x+3), (x-2), (3x+7), and (x+5) – we were able to determine the overall sign of f(x) in each interval. This process involved considering the sign changes that occur at the zeros of each factor. When we focused on the specific interval (-7/3, -3/5), we carefully examined the sign of each factor within this range. We found that (5x+3) is negative, (x-2) is negative, (3x+7) is positive, and (x+5) is positive. By multiplying these signs together, we arrived at the conclusion that f(x) is positive within the interval (-7/3, -3/5). This result is a direct consequence of the fact that there are two negative factors and two positive factors in this interval. The product of an even number of negative factors is positive, which leads to the positive sign of f(x). The analysis we have conducted highlights the importance of understanding the relationship between the zeros of a polynomial function and its sign. The zeros act as critical points where the function's sign can change, and by analyzing the factors of the function, we can systematically determine its sign in different intervals. This technique is not only valuable for solving specific problems, such as determining the sign of f(x) in a given interval, but also for gaining a deeper understanding of the behavior of polynomial functions in general. Moreover, the concepts and techniques we have explored extend beyond the realm of polynomial functions. The idea of analyzing the sign of factors and using a sign chart can be applied to other types of functions as well, such as rational functions. The ability to determine the sign of a function is essential for various mathematical applications, including solving inequalities, sketching graphs, and analyzing the behavior of functions in calculus. In conclusion, our analysis has definitively shown that the sign of f(x) = (5x+3)(x-2)(3x+7)(x+5) is positive in the interval (-7/3, -3/5). This result is a testament to the power of systematic analysis and the importance of understanding the fundamental properties of polynomial functions. This detailed exploration provides a solid foundation for further studies in mathematics and its applications.