Evaluating Polynomial Functions F(x) = X^4 + 6x^3 + 3 At X = 2

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Introduction

In this article, we will delve into the process of evaluating a polynomial function for a specific value of the variable. Polynomial functions are fundamental in mathematics and have wide applications in various fields such as engineering, physics, and computer science. Understanding how to evaluate these functions is crucial for solving equations, graphing curves, and modeling real-world phenomena. The function we will be working with is a polynomial function of degree 4, given by F(x) = x^4 + 6x^3 + 3. Our task is to find the value of this function when x = 2. This involves substituting 2 for every instance of x in the function and then performing the arithmetic operations to simplify the expression. This exercise not only reinforces basic algebraic skills but also highlights the importance of order of operations (PEMDAS/BODMAS) in mathematical calculations. We will explore each step in detail to ensure a clear understanding of the evaluation process. Accurately evaluating functions is a cornerstone of mathematical proficiency, and this article aims to provide a comprehensive guide to achieving that.

Understanding the Function

Before we dive into the evaluation process, it's essential to understand the structure of the function F(x) = x^4 + 6x^3 + 3. This function is a polynomial, which means it's an expression consisting of variables (in this case, x) raised to non-negative integer powers, combined using addition, subtraction, and multiplication, and multiplied by coefficients. The terms of the function are:

  1. x^4: This is the leading term, where the variable x is raised to the power of 4. The coefficient of this term is 1.
  2. 6x^3: Here, the variable x is raised to the power of 3, and it is multiplied by the coefficient 6.
  3. 3: This is a constant term, meaning it does not involve the variable x. It can be thought of as 3x^0 since any number raised to the power of 0 is 1.

The degree of the polynomial is the highest power of the variable in the function, which in this case is 4. Understanding the degree and the coefficients helps in predicting the behavior of the function and its graph. When we evaluate the function for a specific value of x, we are essentially finding the y-coordinate of the point on the graph of the function where the x-coordinate is the given value. In our case, we want to find the y-coordinate when x is 2. This process involves substituting the value of x into the function and simplifying the expression using the correct order of operations. Let's proceed step by step to ensure accuracy.

Step-by-Step Evaluation

To evaluate the function F(x) = x^4 + 6x^3 + 3 for x = 2, we will follow these steps meticulously:

  1. Substitution: Replace every instance of x in the function with the value 2. This gives us:

    F(2) = (2)^4 + 6(2)^3 + 3

    It's crucial to use parentheses when substituting to avoid confusion, especially when dealing with negative numbers or exponents.

  2. Exponents: Next, we evaluate the exponential terms. Remember, the order of operations dictates that exponents should be dealt with before multiplication and addition.

    • (2)^4 = 2 * 2 * 2 * 2 = 16
    • (2)^3 = 2 * 2 * 2 = 8

    Now, substitute these values back into the expression:

    F(2) = 16 + 6(8) + 3

  3. Multiplication: Perform the multiplication operation. We have 6 multiplied by 8:

    • 6 * 8 = 48

    Substitute this result back into the expression:

    F(2) = 16 + 48 + 3

  4. Addition: Finally, we perform the addition operations from left to right:

    • 16 + 48 = 64
    • 64 + 3 = 67

    Therefore, the final result is:

    F(2) = 67

    This step-by-step approach ensures accuracy and clarity in the evaluation process. Each operation is performed in the correct order, leading to the correct result. In the next section, we will verify our answer and discuss common mistakes to avoid.

Verification and Common Mistakes

To ensure the accuracy of our result, it's always a good practice to verify the solution. We have calculated that F(2) = 67 for the function F(x) = x^4 + 6x^3 + 3. Let's quickly recap our steps:

  1. Substitution: F(2) = (2)^4 + 6(2)^3 + 3
  2. Exponents: F(2) = 16 + 6(8) + 3
  3. Multiplication: F(2) = 16 + 48 + 3
  4. Addition: F(2) = 67

The steps are clear and logical, and the arithmetic appears correct. However, it's beneficial to double-check, especially in exams or important calculations. One way to verify is to use a calculator or computational tool to evaluate the function directly. Inputting the expression _(2)^4 + 6(2)^3 + 3_* into a calculator should indeed yield 67, confirming our solution.

Now, let's discuss some common mistakes people make when evaluating functions, particularly polynomial functions:

  1. Incorrect Order of Operations: This is the most frequent error. Forgetting to prioritize exponents before multiplication or addition can lead to a wrong answer. Always remember the PEMDAS/BODMAS rule.
  2. Substitution Errors: Misplacing the value being substituted or failing to substitute in every instance of the variable can lead to significant errors. Always double-check the substitution step.
  3. Arithmetic Errors: Simple addition, subtraction, multiplication, or division errors can occur, especially under time pressure. Taking a moment to carefully perform each operation can prevent these mistakes.
  4. Sign Errors: When dealing with negative numbers, sign errors are common. Pay close attention to the signs when substituting and performing operations.

By being aware of these common pitfalls and practicing careful, methodical evaluation, you can minimize errors and build confidence in your mathematical skills. In the conclusion, we will summarize the key steps and highlight the importance of function evaluation in mathematics.

Conclusion

In this article, we have successfully evaluated the polynomial function F(x) = x^4 + 6x^3 + 3 for x = 2. By following a step-by-step approach, we determined that F(2) = 67. We began by understanding the structure of the function, which is a polynomial of degree 4, and then methodically substituted the value of x into the function. The key steps included:

  1. Substituting x = 2 into the function: F(2) = (2)^4 + 6(2)^3 + 3
  2. Evaluating the exponents: (2)^4 = 16 and (2)^3 = 8
  3. Performing the multiplication: 6 * 8 = 48
  4. Completing the addition: 16 + 48 + 3 = 67

We also emphasized the importance of adhering to the order of operations (PEMDAS/BODMAS) to ensure accurate calculations. Furthermore, we discussed common mistakes that individuals often make when evaluating functions, such as incorrect order of operations, substitution errors, arithmetic errors, and sign errors. By recognizing these potential pitfalls, one can adopt a more cautious and deliberate approach, minimizing the likelihood of errors.

Function evaluation is a fundamental skill in mathematics with widespread applications. It forms the basis for solving equations, graphing functions, and modeling real-world scenarios. Whether you are a student learning algebra or a professional applying mathematical concepts in your field, the ability to accurately evaluate functions is essential. This exercise not only reinforces algebraic manipulation skills but also enhances problem-solving abilities. By practicing and mastering these skills, you can build a strong foundation in mathematics and confidently tackle more complex problems. The process of function evaluation, as demonstrated in this article, is a building block for further mathematical studies and practical applications.

Final Answer: The final answer is (A)