Evaluating Polynomials Finding F(-5) For F(x) = X^2 - 2x - 7

In the realm of mathematics, polynomial functions play a crucial role, serving as the building blocks for numerous mathematical models and real-world applications. Understanding how to evaluate these functions for specific input values is a fundamental skill. This article delves into the process of evaluating a polynomial function, using the example of F(x) = x^2 - 2x - 7 to find F(-5). We will explore the step-by-step method, highlighting the key concepts and calculations involved. This comprehensive guide aims to provide clarity and understanding, empowering you to confidently tackle similar problems.

Understanding Polynomial Functions

Polynomial functions are expressions involving variables raised to non-negative integer powers, combined with coefficients. They are versatile tools in mathematics, used to model various phenomena, from the trajectory of a projectile to the growth of a population. The general form of a polynomial function is:

F(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0

where:

• x is the variable
• a_n, a_{n-1}, ..., a_1, a_0 are the coefficients (constants)
• n is a non-negative integer representing the degree of the polynomial

In our specific example, F(x) = x^2 - 2x - 7, we have a quadratic polynomial (degree 2) with coefficients 1, -2, and -7. The main goal here is to find the value of the function when x is -5. Substituting this value into the equation is the key to solving this problem. Before we delve into the calculations, it's important to understand the concept of function evaluation. Function evaluation is the process of replacing the variable in a function with a specific value and then simplifying the expression. This process allows us to determine the output of the function for a given input. Mastering this skill is essential for understanding the behavior of functions and their applications in various fields.

Step-by-Step Evaluation of F(-5)

To find F(-5) for the polynomial function F(x) = x^2 - 2x - 7, we will follow these steps:

Step 1: Substitute -5 for x

Replace every instance of 'x' in the function with '-5'. Make sure to enclose the substituted value in parentheses, especially when dealing with negative numbers, to avoid any confusion with the order of operations. This gives us:

F(-5) = (-5)^2 - 2(-5) - 7

This substitution is a crucial step as it sets the stage for the rest of the calculation. It's where we transition from a general function to a specific numerical evaluation. The parentheses are particularly important in this step because they ensure that the negative sign is correctly applied during the exponentiation and multiplication operations. Without the parentheses, the calculation might be misinterpreted, leading to an incorrect answer. Pay close attention to this detail to maintain accuracy throughout the process.

Step 2: Evaluate the Exponent

Calculate (-5)^2, which means -5 multiplied by itself. Remember that a negative number multiplied by a negative number results in a positive number:

(-5)^2 = (-5) * (-5) = 25

Now our expression looks like this:

F(-5) = 25 - 2(-5) - 7

Understanding exponents is fundamental in mathematics. In this case, (-5)^2 represents the square of -5. The correct evaluation of this exponent is essential for arriving at the right answer. A common mistake is to treat (-5)^2 as -(5^2), which would result in -25, an incorrect value. The parentheses clearly indicate that the negative sign is part of the base being squared. This emphasizes the importance of adhering to the order of operations (PEMDAS/BODMAS), where exponents are evaluated before multiplication and subtraction. This step solidifies our understanding of how to handle exponents in the context of polynomial functions.

Step 3: Perform the Multiplication

Next, we perform the multiplication -2 * (-5). Again, the product of two negative numbers is a positive number:

-2 * (-5) = 10

Our expression now becomes:

F(-5) = 25 + 10 - 7

Multiplication is a key arithmetic operation in evaluating polynomial functions. In this step, we are multiplying a coefficient (-2) by the substituted value (-5). The sign of the result is determined by the signs of the numbers being multiplied. The rule that the product of two negative numbers is positive is crucial here. This step highlights the importance of being comfortable with signed number arithmetic. By correctly performing the multiplication, we simplify the expression further, bringing us closer to the final answer. Accuracy in this step is paramount to ensuring the overall correctness of the solution. Mastering the rules of sign manipulation is a fundamental skill in algebra and is essential for success in more advanced mathematical concepts.

Step 4: Perform the Addition and Subtraction

Finally, we perform the addition and subtraction from left to right:

25 + 10 = 35

35 - 7 = 28

Therefore, F(-5) = 28

Addition and subtraction are the final steps in evaluating the polynomial function. We perform these operations in the order they appear from left to right, following the standard order of operations. First, we add 25 and 10 to get 35. Then, we subtract 7 from 35, which gives us the final result of 28. This final calculation provides the value of the function F(x) at x = -5. The result, 28, represents the output of the function for the given input. This step emphasizes the importance of careful arithmetic and attention to detail. By performing the addition and subtraction accurately, we arrive at the correct value of F(-5), demonstrating our understanding of function evaluation.

The Answer

So, F(-5) = 28. The correct answer is D. 28.

Conclusion

Evaluating polynomial functions is a fundamental skill in mathematics. By following a step-by-step approach, as demonstrated in this article, you can confidently find the value of a polynomial function for any given input. Remember the importance of substitution, exponent evaluation, multiplication, and addition/subtraction. With practice, you can master this skill and apply it to more complex mathematical problems. Understanding these steps is crucial for solving not only mathematical problems but also real-world problems that can be modeled using polynomial functions. This example provides a solid foundation for further exploration of polynomial functions and their applications.

By carefully substituting the value, following the order of operations, and performing the calculations accurately, we successfully found F(-5). This process highlights the importance of attention to detail and a solid understanding of mathematical principles. This skill is not only essential for academic success but also for various practical applications in science, engineering, and other fields. Continuing to practice these techniques will solidify your understanding and improve your ability to solve more complex mathematical problems. The ability to evaluate polynomial functions is a cornerstone of mathematical literacy and opens the door to a deeper understanding of mathematical concepts.