Subtracting Polynomials A Step By Step Guide To Finding (f-g)(x)

In mathematics, particularly in algebra, polynomial subtraction is a fundamental operation. It involves finding the difference between two polynomials. Polynomials are algebraic expressions consisting of variables and coefficients, combined using addition, subtraction, and multiplication, with non-negative integer exponents. Mastering polynomial subtraction is crucial for solving more complex algebraic problems and understanding higher-level mathematical concepts. This article delves into the process of subtracting two given polynomials, $f(x) = 5x^3 + 8x^2 - 7x - 3$ and $g(x) = 2x^2 - 5x - 10$, to find $(f-g)(x)$. We will break down the steps, provide explanations, and offer insights to help you grasp the concept thoroughly.

Polynomial subtraction is a core concept in algebra, acting as a building block for more advanced mathematical topics. Understanding how to subtract polynomials is essential for various applications, including solving equations, graphing functions, and simplifying expressions. Before diving into the specific problem, let's first establish a solid understanding of the basics. A polynomial is an expression consisting of variables (also called indeterminates) and coefficients, which involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. Examples of polynomials include $3x^2 + 2x - 1$, $5x^4 - 7x^2 + 3$, and $x^3 + 1$. Each term in a polynomial consists of a coefficient and a variable raised to a power. For example, in the term $3x^2$, 3 is the coefficient and $x^2$ is the variable part. When subtracting polynomials, we essentially combine like terms. 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 $2x^3$ are not. The process of polynomial subtraction involves distributing the negative sign to the second polynomial and then combining like terms. This ensures that the correct signs are applied to each term, leading to an accurate result. With a clear understanding of these basics, we can now move on to the problem at hand: finding $(f-g)(x)$ for the given polynomials $f(x)$ and $g(x)$.

Before we proceed with the subtraction, let's clearly define the polynomials we will be working with. We are given two polynomials:

• $f(x) = 5x^3 + 8x^2 - 7x - 3$
• $g(x) = 2x^2 - 5x - 10$

Here, $f(x)$ is a cubic polynomial (degree 3) and $g(x)$ is a quadratic polynomial (degree 2). The degree of a polynomial is the highest power of the variable in the polynomial. Identifying the degree and terms of each polynomial is crucial for performing operations like subtraction correctly. Each term in the polynomials consists of a coefficient and a variable raised to a power. For example, in $f(x)$, the terms are $5x^3$, $8x^2$, $-7x$, and $-3$. Similarly, in $g(x)$, the terms are $2x^2$, $-5x$, and $-10$. The coefficients are the numerical values multiplying the variables, and the powers indicate the degree of each term. When subtracting polynomials, we focus on combining like terms, which are terms with the same variable and power. Therefore, it is essential to accurately identify the terms and their corresponding coefficients and powers in each polynomial.

Understanding the structure of these polynomials sets the stage for the subtraction process. We will carefully align like terms and apply the subtraction operation, ensuring that we combine the coefficients correctly. This step-by-step approach will help us arrive at the correct result for $(f-g)(x)$. Now that we have a clear understanding of the given polynomials, we can move on to the next step: setting up the subtraction operation.

To find $(f-g)(x)$, we need to subtract the polynomial $g(x)$ from the polynomial $f(x)$. This means we will write the expression as:

$(f-g)(x) = f(x) - g(x)$

Now, substitute the given polynomials into this expression:

$(f-g)(x) = (5x^3 + 8x^2 - 7x - 3) - (2x^2 - 5x - 10)$

The next crucial step is to distribute the negative sign in front of the second polynomial, $g(x)$. This means we will multiply each term inside the parentheses of $g(x)$ by -1. Distributing the negative sign correctly is essential for accurate polynomial subtraction. Failing to do so can lead to incorrect results. The negative sign effectively changes the sign of each term in the polynomial being subtracted. For instance, $+2x^2$ becomes $-2x^2$, $-5x$ becomes $+5x$, and $-10$ becomes $+10$. This process is similar to subtracting a negative number, which is equivalent to adding its positive counterpart. After distributing the negative sign, we will have a new expression where we can combine like terms more easily. Aligning like terms can also help visualize the subtraction process. We can write the polynomials vertically, aligning terms with the same powers of $x$ in columns. This method makes it easier to see which terms need to be combined. Once the negative sign is distributed and the terms are aligned, we are ready to perform the subtraction by combining like terms. This step involves adding or subtracting the coefficients of terms with the same variable and power. By carefully setting up the subtraction and distributing the negative sign, we ensure that the subsequent steps will lead to the correct result.

The next crucial step is distributing the negative sign in front of the polynomial $g(x)$. This is done by multiplying each term inside the parentheses of $g(x)$ by -1. Let's perform this operation:

$(f-g)(x) = (5x^3 + 8x^2 - 7x - 3) - (2x^2 - 5x - 10)$

Distribute the negative sign:

$(f-g)(x) = 5x^3 + 8x^2 - 7x - 3 - 2x^2 + 5x + 10$

This step is crucial because it changes the signs of the terms in $g(x)$, preparing them to be combined with the terms in $f(x)$. Distributing the negative sign correctly is essential for accurate polynomial subtraction. Failing to do so can lead to incorrect results. The negative sign effectively changes the operation from subtraction to addition of the negative of the polynomial. For instance, subtracting $2x^2$ becomes adding $-2x^2$, subtracting $-5x$ becomes adding $+5x$, and subtracting $-10$ becomes adding $+10$. This transformation is key to combining like terms properly. After distributing the negative sign, we have a new expression where we can clearly see the terms that need to be combined. The next step involves identifying and combining these like terms to simplify the expression. By carefully distributing the negative sign, we ensure that the subsequent steps will lead to the correct result for $(f-g)(x)$. This step sets the stage for the final simplification, where we will combine like terms to obtain the resulting polynomial.

Now that we have distributed the negative sign, we can combine the like terms in the expression:

$(f-g)(x) = 5x^3 + 8x^2 - 7x - 3 - 2x^2 + 5x + 10$

Identify and combine the like terms:

• $x^3$ terms: $5x^3$ (no other $x^3$ term)
• $x^2$ terms: $8x^2 - 2x^2 = 6x^2$
• $x$ terms: $-7x + 5x = -2x$
• Constant terms: $-3 + 10 = 7$

Combining these terms, we get:

$(f-g)(x) = 5x^3 + 6x^2 - 2x + 7$

The process of combining like terms involves adding or subtracting the coefficients of terms with the same variable and power. This step is crucial for simplifying the expression and obtaining the final result. Like terms are terms that have the same variable raised to the same power. For instance, $8x^2$ and $-2x^2$ are like terms because they both have $x^2$. Similarly, $-7x$ and $5x$ are like terms because they both have $x$. To combine like terms, we add or subtract their coefficients while keeping the variable part the same. For example, $8x^2 - 2x^2$ becomes $(8-2)x^2 = 6x^2$. Constant terms are also like terms, and we simply add or subtract them as well. In this case, $-3 + 10 = 7$. The $5x^3$ term has no like terms, so it remains unchanged in the final expression. By carefully combining like terms, we simplify the expression and arrive at the final polynomial. This step demonstrates the importance of accurate arithmetic and attention to detail in polynomial operations. With the like terms combined, we have successfully found the expression for $(f-g)(x)$.

After combining the like terms, we have found the result of the subtraction:

$(f-g)(x) = 5x^3 + 6x^2 - 2x + 7$

This is the polynomial obtained by subtracting $g(x)$ from $f(x)$. The result is a cubic polynomial, which is the same degree as the higher-degree polynomial, $f(x)$. The polynomial $5x^3 + 6x^2 - 2x + 7$ represents the difference between the two original polynomials. Each term in the resulting polynomial has a specific coefficient and power of $x$. The coefficients indicate the magnitude of each term, and the powers determine the degree of each term. The constant term, 7, represents the value of the polynomial when $x$ is zero. This resulting polynomial can be used in further algebraic operations, such as solving equations, graphing functions, or evaluating the polynomial for specific values of $x$. Understanding the structure and properties of the resulting polynomial is crucial for applying it in various mathematical contexts. By successfully performing the subtraction and obtaining the result, we have demonstrated the process of polynomial subtraction. This skill is fundamental in algebra and is used in many areas of mathematics and its applications.

Understanding polynomial subtraction is not just an academic exercise; it has numerous practical applications in various fields. In engineering, polynomial subtraction can be used in circuit analysis, control systems, and signal processing. In computer graphics, it is used in curve fitting and surface modeling. In economics, it can be used in cost analysis and optimization problems. Furthermore, polynomial subtraction is a building block for more advanced mathematical concepts, such as polynomial division, factoring, and solving polynomial equations. Exploring these related topics can deepen your understanding of algebra and its applications. Polynomial division, for example, is the reverse operation of polynomial multiplication and is used to simplify rational expressions and solve equations. Factoring polynomials involves expressing a polynomial as a product of simpler polynomials, which is crucial for solving polynomial equations. Solving polynomial equations is a fundamental skill in algebra and has applications in many areas of science and engineering. By mastering polynomial subtraction and related concepts, you can tackle a wide range of mathematical problems and gain a deeper appreciation for the power and versatility of algebra. This foundational understanding will serve you well in more advanced studies and in practical applications across various disciplines. Further exploration of these concepts will enhance your mathematical toolkit and problem-solving abilities.

In summary, we have successfully found $(f-g)(x)$ for the given polynomials $f(x) = 5x^3 + 8x^2 - 7x - 3$ and $g(x) = 2x^2 - 5x - 10$. The process involved setting up the subtraction, distributing the negative sign, combining like terms, and arriving at the result:

$(f-g)(x) = 5x^3 + 6x^2 - 2x + 7$

This exercise demonstrates the fundamental principles of polynomial subtraction. Polynomial subtraction is a core skill in algebra and is essential for solving various mathematical problems. By understanding the steps involved, you can confidently subtract polynomials and apply this knowledge in more complex scenarios. The key to success lies in accurately distributing the negative sign and carefully combining like terms. Attention to detail and practice are crucial for mastering this skill. Moreover, understanding polynomial subtraction opens the door to exploring related concepts, such as polynomial division, factoring, and solving polynomial equations. These concepts build upon the foundation of polynomial subtraction and further enhance your algebraic skills. As you continue your mathematical journey, mastering these fundamental operations will prove invaluable in tackling more advanced topics and real-world applications. Remember, practice makes perfect, so continue to work through examples and explore different types of polynomial subtraction problems to solidify your understanding.