Finding (f/g)(x) And (f/g)(-4) For Polynomial Functions

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In the realm of mathematical functions, the quotient of two functions, denoted as (f/g)(x), plays a crucial role in various applications. This article delves into the intricacies of finding the quotient function and evaluating it at a specific point, using the example functions f(x) = 4x⁴ - 20x³ + 9x² - 54x + 45 and g(x) = x - 5. We will explore the process of polynomial division to determine the quotient function (f/g)(x) and then substitute x = -4 to find (f/g)(-4). This comprehensive guide aims to provide a clear understanding of the concepts and techniques involved, enabling you to confidently tackle similar problems.

The quotient function (f/g)(x) is defined as the division of two functions, f(x) and g(x), where g(x) ≠ 0. In this case, we have f(x) = 4x⁴ - 20x³ + 9x² - 54x + 45 and g(x) = x - 5. To find (f/g)(x), we need to perform polynomial division.

Polynomial division is a method for dividing a polynomial by another polynomial of a lower or equal degree. It is similar to the long division method used for dividing numbers. The process involves dividing the leading term of the dividend (f(x)) by the leading term of the divisor (g(x)), multiplying the result by the divisor, subtracting it from the dividend, and bringing down the next term. This process is repeated until the degree of the remainder is less than the degree of the divisor.

Let's perform the polynomial division for our given functions:

 4x³ + 9x + 9
 x - 5 | 4x⁴ - 20x³ + 9x² - 54x + 45
 - (4x⁴ - 20x³)
 ------------------
 9x² - 54x
 - (9x² - 45x)
 ------------------
 -9x + 45
 - (-9x + 45)
 ------------------
 0

From the polynomial division, we find that the quotient is 4x³ + 9x + 9 and the remainder is 0. Therefore, the quotient function (f/g)(x) is:

(f/g)(x) = 4x³ + 9x + 9

This resulting polynomial represents the function obtained after dividing f(x) by g(x). It's a crucial step in understanding the relationship between the two original functions and forms the basis for further analysis.

Step-by-Step Breakdown of Polynomial Division

To ensure a clear understanding of the process, let's break down the polynomial division step by step:

  1. Set up the division: Write the dividend (4x⁴ - 20x³ + 9x² - 54x + 45) inside the division symbol and the divisor (x - 5) outside.
  2. Divide the leading terms: Divide the leading term of the dividend (4x⁴) by the leading term of the divisor (x), which gives 4x³.
  3. Multiply the quotient by the divisor: Multiply the result (4x³) by the divisor (x - 5), which gives 4x⁴ - 20x³.
  4. Subtract from the dividend: Subtract the result (4x⁴ - 20x³) from the corresponding terms in the dividend. This eliminates the leading terms.
  5. Bring down the next term: Bring down the next term from the dividend (9x²).
  6. Repeat the process: Repeat steps 2-5 with the new polynomial (9x² - 54x). Divide the leading term (9x²) by the leading term of the divisor (x), which gives 9x. Multiply 9x by (x-5) to get 9x² - 45x. Subtract this from 9x² - 54x to get -9x. Bring down the next term (+45).
  7. Continue until the remainder's degree is less than the divisor's: Repeat the process again. Divide -9x by x to get -9. Multiply -9 by (x-5) to get -9x + 45. Subtracting this from -9x + 45 gives a remainder of 0.
  8. Identify the quotient and remainder: The quotient is the polynomial obtained above the division symbol (4x³ + 9x - 9), and the remainder is the polynomial left at the end (0).

Understanding each of these steps is crucial for mastering polynomial division. Practice with different examples will solidify your understanding and improve your proficiency in this technique.

Now that we have found the quotient function (f/g)(x) = 4x³ + 9x + 9, we can evaluate it at x = -4. This involves substituting -4 for x in the quotient function and simplifying the expression.

(f/g)(-4) = 4(-4)³ + 9(-4) + 9

Let's simplify this expression step by step:

  1. Calculate (-4)³: (-4)³ = -4 * -4 * -4 = -64
  2. Multiply 4 by (-64): 4 * (-64) = -256
  3. Multiply 9 by (-4): 9 * (-4) = -36
  4. Substitute the values: (f/g)(-4) = -256 - 36 + 9
  5. Simplify the expression: (f/g)(-4) = -292 + 9 = -283

Therefore, the value of the quotient function (f/g)(x) at x = -4 is:

(f/g)(-4) = -283

This result signifies the value of the function resulting from the division of f(x) by g(x) when x is specifically -4. It's a numerical representation of the function's behavior at that point.

Significance of Evaluating Functions

Evaluating functions at specific points is a fundamental concept in mathematics with wide-ranging applications. It allows us to:

  • Determine the function's output for a given input: This is the most basic application. By substituting a value for x, we find the corresponding y-value on the function's graph.
  • Analyze the function's behavior: Evaluating a function at different points can reveal trends, such as increasing or decreasing intervals, maximum and minimum values, and points of inflection.
  • Solve equations: Finding the x-values where a function equals a certain value (e.g., finding the roots) often involves evaluating the function.
  • Model real-world phenomena: Functions are used to model various phenomena in science, engineering, and economics. Evaluating these functions helps us make predictions and understand the behavior of the modeled system.

In this specific case, evaluating (f/g)(-4) gives us a numerical value for the relationship between f(x) and g(x) when x is -4. This could be useful in a variety of contexts, depending on what f(x) and g(x) represent.

In this article, we have thoroughly explored the process of finding the quotient function (f/g)(x) for the given functions f(x) = 4x⁴ - 20x³ + 9x² - 54x + 45 and g(x) = x - 5. We successfully performed polynomial division to obtain (f/g)(x) = 4x³ + 9x + 9. Furthermore, we evaluated the quotient function at x = -4, finding that (f/g)(-4) = -283. This exploration provides a solid foundation for understanding and working with quotient functions and their evaluations.

The concepts and techniques discussed in this article are fundamental in mathematics and have applications in various fields. Mastering polynomial division and function evaluation is crucial for solving more complex problems and understanding mathematical models. By practicing these techniques with different examples, you can enhance your mathematical skills and confidently tackle a wide range of problems involving functions and their operations. The ability to manipulate and analyze functions is a key skill in mathematics, science, and engineering, and this guide provides a comprehensive starting point for developing that skill.