Subtracting Algebraic Expressions A = (10x⁴ - X³ - 3x² + X + 1) - (x³ - 5x + 7)

Calculate the subtraction of the algebraic expressions A = (10x⁴ - x³ - 3x² + x + 1) - (x³ - 5x + 7).

In the realm of algebra, subtracting expressions is a fundamental operation. It's more than just manipulating symbols; it's about understanding the underlying structure of polynomials and how they interact. In this comprehensive guide, we will explore the intricacies of subtracting algebraic expressions, using the specific example of (10x⁴ - x³ - 3x² + x + 1) - (x³ - 5x + 7) as our central focus. This detailed analysis will not only provide a step-by-step solution but also delve into the reasoning behind each step, ensuring a solid grasp of the concepts involved.

Understanding the Basics of Algebraic Expressions

Before diving into the subtraction process, it's crucial to establish a strong foundation in the basics of algebraic expressions. An algebraic expression is a combination of variables, constants, and mathematical operations. Variables are symbols (usually letters) that represent unknown values, while constants are fixed numerical values. These components are connected by operations such as addition, subtraction, multiplication, and division.

Polynomials are a specific type of algebraic expression that play a central role in algebra. They are defined as expressions consisting of variables raised to non-negative integer powers, combined using addition, subtraction, and multiplication. The expression 10x⁴ - x³ - 3x² + x + 1 is a polynomial, where each term consists of a coefficient (the numerical factor) and a variable raised to a power.

Key Components of a Polynomial

• Terms: Each part of a polynomial separated by addition or subtraction is called a term. In our example, the terms are 10x⁴, -x³, -3x², x, and 1.
• Coefficients: The numerical factor of each term is the coefficient. The coefficients in our example are 10, -1, -3, 1, and 1.
• Variables: The variables are the symbols representing unknown values, which is 'x' in our case.
• Exponents: The exponents indicate the power to which the variable is raised. The exponents in our example are 4, 3, 2, 1 (for the term 'x'), and 0 (for the constant term 1, since x⁰ = 1).
• Degree: The highest exponent of the variable in the polynomial is the degree of the polynomial. The degree of 10x⁴ - x³ - 3x² + x + 1 is 4.

Understanding these components is essential for performing operations on polynomials, including subtraction. When subtracting algebraic expressions, we are essentially combining like terms. Like terms are terms that have the same variable raised to the same power. For example, 3x² and -5x² are like terms, while 3x² and 3x³ are not.

Step-by-Step Subtraction of (10x⁴ - x³ - 3x² + x + 1) - (x³ - 5x + 7)

Now, let's dive into the process of subtracting the algebraic expressions (10x⁴ - x³ - 3x² + x + 1) - (x³ - 5x + 7). We'll break down each step to ensure clarity and understanding.

Step 1: Distribute the Negative Sign

The first critical step in subtracting algebraic expressions is to distribute the negative sign (or minus sign) to each term within the second expression. This is equivalent to multiplying each term inside the parentheses by -1. It's a crucial step because it correctly changes the signs of the terms being subtracted.

So, (10x⁴ - x³ - 3x² + x + 1) - (x³ - 5x + 7) becomes 10x⁴ - x³ - 3x² + x + 1 - x³ + 5x - 7.

Notice how the signs of the terms in the second expression have changed: +x³ became -x³, -5x became +5x, and +7 became -7. This sign change is essential for accurate subtraction.

Step 2: Identify and Group Like Terms

After distributing the negative sign, the next step is to identify and group like terms. Remember, like terms are those with the same variable raised to the same power. Grouping them together makes the next step of combining them much easier and reduces the chance of errors.

In our expression 10x⁴ - x³ - 3x² + x + 1 - x³ + 5x - 7, the like terms are:

• x⁴ terms: 10x⁴ (There's only one term with x⁴)
• x³ terms: -x³ and -x³
• x² terms: -3x² (There's only one term with x²)
• x terms: x and 5x
• Constant terms: 1 and -7

We can rearrange the expression to visually group the like terms together:

10x⁴ - x³ - x³ - 3x² + x + 5x + 1 - 7

This grouping helps to clearly see which terms can be combined in the next step.

Step 3: Combine Like Terms

Now, with the like terms grouped together, we can combine them by adding or subtracting their coefficients. This is the core of the subtraction process, where we simplify the expression by reducing the number of terms.

Let's combine the like terms in our expression:

• x⁴ terms: 10x⁴ (remains as is since there are no other x⁴ terms)
• x³ terms: -x³ - x³ = -2x³ (We add the coefficients -1 and -1)
• x² terms: -3x² (remains as is since there are no other x² terms)
• x terms: x + 5x = 6x (We add the coefficients 1 and 5)
• Constant terms: 1 - 7 = -6 (We subtract 7 from 1)

Step 4: Write the Simplified Expression

Finally, we write the simplified expression by combining the results from the previous step. This gives us the final answer after subtracting the two algebraic expressions.

Combining the results, we get:

10x⁴ - 2x³ - 3x² + 6x - 6

This is the result of subtracting (x³ - 5x + 7) from (10x⁴ - x³ - 3x² + x + 1). The expression is now in its simplest form, with all like terms combined.

Best Practices for Subtracting Algebraic Expressions

To ensure accuracy and efficiency when subtracting algebraic expressions, consider these best practices:

1. Distribute the Negative Sign Carefully: This is the most common area for errors. Double-check that you have correctly changed the signs of all terms in the second expression.
2. Organize Your Work: Write neatly and align like terms vertically or use color-coding to keep them organized. This helps prevent mistakes when combining terms.
3. Double-Check Your Answer: After simplifying, review your work to ensure you haven't missed any terms or made any sign errors. You can also substitute a value for the variable (e.g., x = 1) into both the original expression and the simplified expression to verify they are equivalent.
4. Practice Regularly: Like any mathematical skill, proficiency in subtracting algebraic expressions comes with practice. Work through various examples to build your confidence and accuracy.

Common Mistakes to Avoid

• Forgetting to Distribute the Negative Sign: This leads to incorrect signs in the simplified expression.
• Combining Unlike Terms: Only like terms can be combined. Make sure the variables and their exponents are the same before combining terms.
• Arithmetic Errors: Be careful when adding and subtracting coefficients, especially when dealing with negative numbers.
• Skipping Steps: Avoid rushing through the process. Write out each step to minimize errors.

Real-World Applications of Subtracting Algebraic Expressions

Subtracting algebraic expressions isn't just an abstract mathematical concept; it has practical applications in various fields. Here are a few examples:

• Engineering: Engineers use algebraic expressions to model physical systems and perform calculations. Subtracting expressions can help determine the difference in forces, voltages, or other quantities.
• Physics: Physicists use algebraic expressions to describe motion, energy, and other physical phenomena. Subtracting expressions can help calculate changes in these quantities.
• Economics: Economists use algebraic expressions to model supply, demand, and other economic factors. Subtracting expressions can help analyze the impact of different policies or market conditions.
• Computer Science: Programmers use algebraic expressions in algorithms and data structures. Subtracting expressions can be used in optimization problems or to calculate differences between data sets.

Conclusion: Mastering Subtraction of Algebraic Expressions

Subtracting algebraic expressions is a fundamental skill in algebra with wide-ranging applications. By understanding the basics of polynomials, carefully distributing the negative sign, grouping and combining like terms, and following best practices, you can master this operation. The specific example of (10x⁴ - x³ - 3x² + x + 1) - (x³ - 5x + 7) provides a concrete illustration of the process, but the principles apply to any subtraction of algebraic expressions. With practice and attention to detail, you can confidently tackle even more complex algebraic problems.