Finding The Sum Of Polynomial Functions F(x) + G(x)

In mathematics, polynomial functions play a fundamental role, and understanding how to manipulate them is crucial for various applications. One common operation involves finding the sum of two or more polynomial functions. In this article, we will delve into the process of adding polynomial functions, focusing on the specific example where $f(x) = x^2 - 2x - 48$ and $g(x) = x + 6$. Our goal is to determine the sum $f(x) + g(x)$ and express the result in its simplest polynomial form. This exploration will not only enhance your understanding of polynomial operations but also provide a practical approach to solving similar problems. Before we jump into the solution, let's first discuss the basics of polynomial functions and how they are manipulated.

Polynomial functions are algebraic expressions that involve variables raised to non-negative integer powers. They are the cornerstone of algebra and calculus, and their properties are extensively studied in mathematics. A polynomial function can be written in the general form:

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

where $a_n, a_{n-1}, ..., a_1, a_0$ are constants (coefficients), and $n$ is a non-negative integer (the degree of the polynomial). The terms of a polynomial are the individual expressions separated by addition or subtraction signs, and like terms are those that have the same variable raised to the same power. Adding polynomials involves combining like terms, a process that simplifies the expression while preserving its mathematical integrity. This basic operation is essential for more complex manipulations and problem-solving in algebra and calculus. By mastering the addition of polynomials, you lay a solid foundation for tackling a wide range of mathematical challenges. In the following sections, we will apply this principle to the specific functions given, providing a step-by-step solution and explanation.

Before we dive into finding $f(x) + g(x)$, let's briefly discuss polynomial functions. Polynomial functions are expressions that consist of variables raised to non-negative integer powers, combined with coefficients and constants. They are a fundamental concept in algebra and calculus, and understanding their properties is crucial for solving a wide range of mathematical problems. A polynomial function can be written in the general form:

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

where:

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

Like terms are terms that have the same variable raised to the same power. For example, $3x^2$ and $-5x^2$ are like terms, while $2x$ and $2x^2$ are not. When adding polynomials, we combine like terms to simplify the expression. This involves adding or subtracting the coefficients of the like terms while keeping the variable and exponent the same. This process is a direct application of the distributive property and is essential for expressing the sum of polynomials in its simplest form. In essence, adding polynomials is a matter of grouping and combining terms that are algebraically compatible. This skill is not only fundamental for polynomial arithmetic but also for more advanced topics like polynomial factorization and calculus. Therefore, a solid grasp of how to add polynomials forms a cornerstone for further mathematical studies.

We are given two polynomial functions:

• $f(x) = x^2 - 2x - 48$
• $g(x) = x + 6$

The problem asks us to find the sum of these two functions, $f(x) + g(x)$, and express the result as a polynomial in its simplest form. This means we need to combine the two functions by adding their corresponding terms and then simplifying the resulting expression by combining like terms. The simplification process ensures that the polynomial is written in its most concise form, making it easier to analyze and use in further calculations. The challenge lies in correctly identifying and combining the like terms, which involves paying close attention to the powers of $x$ and their respective coefficients. This process is a fundamental skill in algebra and is used extensively in higher-level mathematics. By accurately finding the sum and simplifying it, we can gain a better understanding of the relationship between the two original functions and their combined behavior. The steps involved in this process will be detailed in the following sections, providing a clear and methodical approach to solving the problem.

To find $f(x) + g(x)$, we add the two functions together:

$f(x) + g(x) = (x^2 - 2x - 48) + (x + 6)$

Step 1: Combine the expressions

First, we write out the sum of the two polynomials:

$x^2 - 2x - 48 + x + 6$

Step 2: Identify like terms

Next, we identify the like terms, which are the terms with the same variable raised to the same power. In this expression, we have:

• $x^2$ term: $x^2$
• $x$ terms: $-2x$ and $+x$
• Constant terms: $-48$ and $+6$

Step 3: Combine like terms

Now, we combine the like terms by adding their coefficients:

• $x^2$ term: There is only one $x^2$ term, so it remains $x^2$.
• $x$ terms: $-2x + x = -1x = -x$
• Constant terms: $-48 + 6 = -42$

Step 4: Write the simplified polynomial

Finally, we write the simplified polynomial by combining the results from the previous step:

$f(x) + g(x) = x^2 - x - 42$

This is the sum of the two functions expressed as a polynomial in its simplest form. The process of combining like terms is a fundamental algebraic operation that allows us to express complex expressions in a more manageable and understandable form. By systematically identifying and combining these terms, we can simplify polynomials and make them easier to work with in various mathematical contexts. This step-by-step approach not only provides the solution but also reinforces the importance of methodical problem-solving in mathematics.

Therefore, the sum of the functions $f(x) = x^2 - 2x - 48$ and $g(x) = x + 6$ is:

$f(x) + g(x) = x^2 - x - 42$

This result is a polynomial in simplest form, as there are no more like terms to combine. The process of finding this sum involved carefully identifying and combining the like terms from the original polynomials. This is a fundamental skill in algebra, and mastering it is crucial for success in more advanced mathematical topics. The final polynomial, $x^2 - x - 42$, represents the combined behavior of the two original functions. It can be used for various purposes, such as finding the roots of the combined function or graphing its behavior. Understanding how to manipulate polynomial functions in this way is a key step in developing a strong mathematical foundation. The ability to add, subtract, multiply, and divide polynomials is essential for solving a wide range of problems in algebra, calculus, and other areas of mathematics. This example demonstrates a clear and concise method for adding polynomials, which can be applied to more complex problems as well.

In this article, we successfully found the sum of two polynomial functions, $f(x) = x^2 - 2x - 48$ and $g(x) = x + 6$. We followed a step-by-step process to combine the functions and express the result as a polynomial in simplest form. This involved identifying like terms and combining their coefficients. The final result, $f(x) + g(x) = x^2 - x - 42$, represents the sum of the two functions in a simplified manner. Understanding how to add polynomial functions is a crucial skill in algebra and higher-level mathematics. It forms the basis for more complex operations and problem-solving techniques. By mastering the process of combining like terms and simplifying expressions, you can tackle a wide range of mathematical challenges with confidence. This example provides a clear illustration of how to approach such problems, and the principles discussed can be applied to various other scenarios involving polynomial functions. The ability to manipulate polynomials effectively is essential for students and professionals alike, as these functions are widely used in various fields, including engineering, physics, and computer science. Therefore, a strong understanding of polynomial operations is a valuable asset in any mathematical endeavor.