Polynomial Division And Evaluation Finding (f/g)(x) And (f/g)(0)

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Introduction

In this article, we delve into the fascinating world of polynomial functions and their operations. Our primary focus will be on the functions f(x) = 4x³ - 6x² - 31x - 36 and g(x) = x - 4. We aim to explore the concept of dividing one polynomial function by another, specifically finding (f/g)(x), and then evaluating the resulting function at a particular point, (f/g)(0). This exploration will provide a deeper understanding of polynomial division, function composition, and evaluation, which are fundamental concepts in algebra and calculus. Understanding these concepts is crucial for solving various mathematical problems and real-world applications. We will break down the steps involved in polynomial long division and function evaluation, making it easy to follow and comprehend. Whether you're a student learning about polynomial functions or someone looking to refresh your algebra skills, this article will provide a comprehensive guide.

Finding (f/g)(x)

To determine (f/g)(x), we need to perform polynomial division, dividing the function f(x) by g(x). Polynomial division is a method similar to long division with numbers, but instead of digits, we work with terms containing variables and their coefficients. The function f(x) = 4x³ - 6x² - 31x - 36 is the dividend, and g(x) = x - 4 is the divisor. We set up the long division problem and proceed step by step. First, we focus on the leading terms of both polynomials. We ask ourselves, what do we need to multiply x (the leading term of the divisor) by to get 4x³ (the leading term of the dividend)? The answer is 4x². We then multiply the entire divisor, x - 4, by 4x², which gives us 4x³ - 16x². We subtract this result from the dividend. This process eliminates the leading term of the dividend, and we bring down the next term. We continue this process until we have divided all terms of the dividend. If there is a remainder, we write it as a fraction over the divisor. The quotient we obtain from this division is the simplified form of (f/g)(x). This process may seem complex at first, but with practice, it becomes a straightforward method for dividing polynomials. Understanding polynomial division is essential for simplifying rational expressions, solving polynomial equations, and various other algebraic manipulations. In the next section, we will perform the long division step-by-step to illustrate this process clearly.

Step-by-Step Polynomial Long Division

Let's perform the polynomial long division to find (f/g)(x), where f(x) = 4x³ - 6x² - 31x - 36 and g(x) = x - 4. We set up the long division as follows:

 x - 4 | 4x³ - 6x² - 31x - 36

  1. Divide the leading terms: Divide 4x³ by x, which gives 4x². This is the first term of our quotient.

4x² x - 4 | 4x³ - 6x² - 31x - 36


2.  **Multiply the divisor by the quotient term:** Multiply *(x - 4)* by *4x²*, resulting in *4x³ - 16x²*.

    ```
 4x²
 x - 4 | 4x³ - 6x² - 31x - 36
 4x³ - 16x²

  1. Subtract: Subtract the result from the dividend:

4x² x - 4 | 4x³ - 6x² - 31x - 36 -(4x³ - 16x²)

10x² - 31x


4.  **Bring down the next term:** Bring down the next term from the dividend, which is *-31x*.

    ```
 4x²
 x - 4 | 4x³ - 6x² - 31x - 36
 -(4x³ - 16x²)
 ------------------
 10x² - 31x

  1. Repeat the process: Divide 10x² by x, which gives 10x. This is the next term of our quotient.

4x² + 10x x - 4 | 4x³ - 6x² - 31x - 36 10x² - 31x


6.  **Multiply the divisor by the new quotient term:** Multiply *(x - 4)* by *10x*, resulting in *10x² - 40x*.

    ```
 4x² + 10x
 x - 4 | 4x³ - 6x² - 31x - 36
 10x² - 31x
 10x² - 40x

  1. Subtract: Subtract the result:

4x² + 10x x - 4 | 4x³ - 6x² - 31x - 36 -(10x² - 40x)

9x - 36


8.  **Bring down the next term:** Bring down the last term, which is *-36*.

    ```
 4x² + 10x
 x - 4 | 4x³ - 6x² - 31x - 36
 9x - 36

  1. Repeat the process again: Divide 9x by x, which gives 9. This is the final term of our quotient.

4x² + 10x + 9 x - 4 | 4x³ - 6x² - 31x - 36 9x - 36


10. **Multiply the divisor by the new quotient term:** Multiply *(x - 4)* by *9*, resulting in *9x - 36*.

    ```
 4x² + 10x + 9
 x - 4 | 4x³ - 6x² - 31x - 36
 9x - 36
 9x - 36

  1. Subtract: Subtract the result:

4x² + 10x + 9 x - 4 | 4x³ - 6x² - 31x - 36 -(9x - 36)

0


Since the remainder is 0, the division is exact. Therefore, **(f/g)(x) = 4x² + 10x + 9**. This result is a quadratic function, which is a polynomial of degree 2. Understanding the quotient obtained from polynomial division allows us to analyze the behavior of the original function *f(x)* in relation to the divisor *g(x)*. In the next section, we will evaluate this resulting function at x = 0 to find (f/g)(0).

## Evaluating (f/g)(0)

Now that we have found **(f/g)(x) = 4x² + 10x + 9**, we can evaluate this function at *x = 0*. Evaluating a function at a specific point involves substituting that value for the variable in the function and simplifying the expression. In this case, we will substitute *0* for *x* in the expression *4x² + 10x + 9*. This process is a fundamental concept in mathematics and is used extensively in various fields, including calculus, physics, and engineering. Function evaluation allows us to determine the output of a function for a given input, providing valuable information about the function's behavior and characteristics. In the context of polynomial functions, evaluating at specific points can help us understand the function's graph, find its roots, and solve related problems. By substituting *x = 0*, we can find the y-intercept of the quadratic function (f/g)(x), which is a crucial point for understanding the function's graph and behavior. This evaluation will give us a numerical value, representing the function's output when the input is zero. In the following steps, we will perform the substitution and simplification to find the value of (f/g)(0).

### Step-by-Step Evaluation

To find **(f/g)(0)**, we substitute *x = 0* into the function *(f/g)(x) = 4x² + 10x + 9*:*

1.  **Substitute x = 0:**

    ```
    (f/g)(0) = 4(0)² + 10(0) + 9
    ```

2.  **Simplify the terms:**

    ```
    (f/g)(0) = 4(0) + 10(0) + 9
    ```

3.  **Continue simplifying:**

    ```
    (f/g)(0) = 0 + 0 + 9
    ```

4.  **Final result:**

    ```
    (f/g)(0) = 9
    ```

Therefore, **(f/g)(0) = 9**. This means that when we divide the polynomial function *f(x)* by *g(x)* and then evaluate the resulting function at *x = 0*, the output is 9. This result can be interpreted graphically as the y-intercept of the quadratic function (f/g)(x). The y-intercept is the point where the graph of the function intersects the y-axis, and it provides valuable information about the function's behavior. In this case, the y-intercept is at the point (0, 9). Understanding the value of (f/g)(0) allows us to gain insights into the function's properties and its relationship to the original polynomials *f(x)* and *g(x)*.

## Conclusion

In conclusion, we have successfully found **(f/g)(x) = 4x² + 10x + 9** through polynomial long division and evaluated **(f/g)(0) = 9**. This exercise demonstrates the process of dividing one polynomial by another and evaluating the resulting function at a specific point. Polynomial division is a fundamental algebraic technique with applications in various areas of mathematics, including simplifying rational expressions, solving polynomial equations, and finding asymptotes of rational functions. Function evaluation, on the other hand, is a core concept in understanding the behavior of functions and their graphical representation. By combining these two concepts, we can gain a deeper understanding of the relationship between polynomials and their quotients. The result (f/g)(0) = 9 provides us with a specific point on the graph of the function (f/g)(x), which is the y-intercept. This information is valuable for sketching the graph of the function and understanding its overall behavior. The process we followed in this article can be applied to other polynomial functions and provides a solid foundation for further exploration of algebraic concepts. Understanding polynomial division and function evaluation is essential for success in higher-level mathematics courses and in various fields that rely on mathematical modeling and analysis. The skills and knowledge gained from this exploration will be beneficial in tackling more complex problems involving polynomial functions and their applications.