Simplifying Algebraic Expressions Combining Like Terms And Ordering By Power

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In mathematics, simplifying expressions is a fundamental skill. It allows us to take complex expressions and reduce them to their most basic form, making them easier to understand and work with. One of the key techniques in simplifying expressions is combining like terms. This involves identifying terms that have the same variable raised to the same power and then adding or subtracting their coefficients. By doing so, we can significantly reduce the complexity of an expression.

Understanding Like Terms

Before we delve into the process of combining like terms, it's crucial to understand what exactly constitutes a like term. In algebraic expressions, a term is a single number, a variable, or a product of numbers and variables. Like terms are those that share the same variable raised to the same power. For example, in the expression 3x^2 + 5x - 2x^2 + 7, 3x^2 and -2x^2 are like terms because they both have the variable x raised to the power of 2. On the other hand, 5x is not a like term with 3x^2 or -2x^2 because it has the variable x raised to the power of 1.

To further illustrate this concept, consider the expression 4y^3 - 2y + y^3 + 6y - 1. Here, 4y^3 and y^3 are like terms as they both involve y cubed. Similarly, -2y and 6y are like terms since they both have y to the power of 1. The constant term -1 doesn't have any variable, so it doesn't have any like terms in this expression. Understanding how to identify like terms is the foundation for simplifying expressions effectively.

Combining Like Terms

Once we've identified like terms within an expression, the next step is to combine them. This involves adding or subtracting the coefficients of the like terms while keeping the variable and its exponent the same. The coefficient is the numerical factor that multiplies the variable. For instance, in the term 5x^2, the coefficient is 5.

To combine like terms, we apply the distributive property in reverse. The distributive property states that a(b + c) = ab + ac. When combining like terms, we factor out the common variable and its exponent and then add or subtract the coefficients. Let's consider the expression 3x^2 + 5x - 2x^2 + 7. We identified 3x^2 and -2x^2 as like terms. To combine them, we add their coefficients: 3 + (-2) = 1. Therefore, 3x^2 - 2x^2 simplifies to 1x^2, which is usually written simply as x^2. The expression now becomes x^2 + 5x + 7. Since 5x and 7 are not like terms with x^2, we cannot combine them further.

The process of combining like terms is a direct application of the distributive property and is a cornerstone of algebraic simplification. It's essential to pay close attention to the signs of the coefficients, as they determine whether we add or subtract the terms. By systematically combining like terms, we can transform complex expressions into simpler, more manageable forms.

Ordering Terms by Power

In addition to combining like terms, it's common practice to write the terms of a simplified expression in a specific order. The standard convention is to arrange the terms in descending order of their powers. This means that the term with the highest exponent comes first, followed by the term with the next highest exponent, and so on, until we reach the constant term (if there is one).

For example, consider the expression -7b^2 - 3b + 2b - 2 - 11b^2 - 5 - b^2 + 2b. After combining like terms, we might arrive at an expression like -19b^2 + b - 7. To order the terms by power, we identify the term with the highest exponent, which is -19b^2 (exponent of 2). Next, we have the term b (exponent of 1), and finally, the constant term -7 (exponent of 0). Therefore, the expression is already correctly ordered in this case.

Writing expressions in descending order of powers provides a consistent and organized way to represent polynomials. It makes it easier to compare expressions, identify their degree, and perform further operations. This convention is widely used in mathematics and is crucial for clear communication and accurate calculations.

Step-by-Step Simplification

Now, let's apply these concepts to simplify the given expression: -7b^2 - 3b + 2b - 2 - 11b^2 - 5 - b^2 + 2b. We'll follow a step-by-step approach to ensure accuracy:

  1. Identify Like Terms: In this expression, we have three types of like terms: terms with b^2, terms with b, and constant terms. The b^2 terms are -7b^2, -11b^2, and -b^2. The b terms are -3b, 2b, and 2b. The constant terms are -2 and -5.
  2. Combine Like Terms with b^2: We add the coefficients of the b^2 terms: -7 + (-11) + (-1) = -19. So, -7b^2 - 11b^2 - b^2 simplifies to -19b^2.
  3. Combine Like Terms with b: We add the coefficients of the b terms: -3 + 2 + 2 = 1. So, -3b + 2b + 2b simplifies to 1b, which is usually written as b.
  4. Combine Constant Terms: We add the constant terms: -2 + (-5) = -7. So, -2 - 5 simplifies to -7.
  5. Write the Simplified Expression: Now we combine the simplified terms: -19b^2 + b - 7.

Thus, the simplified expression is -19b^2 + b - 7. We have successfully combined like terms and written the expression in descending order of powers. This step-by-step approach ensures that we accurately simplify expressions, even when they appear complex.

Importance of Simplification

Simplifying expressions is not just an academic exercise; it's a vital skill with numerous applications in mathematics and beyond. Simplified expressions are easier to understand, manipulate, and use in further calculations. In algebra, calculus, and other advanced mathematical fields, simplifying expressions is often a necessary step before solving equations, graphing functions, or performing other operations.

In real-world applications, simplifying expressions can help us model and solve problems in various fields, such as physics, engineering, economics, and computer science. For example, in physics, we might use algebraic expressions to describe the motion of an object. Simplifying these expressions can make it easier to predict the object's position and velocity. In economics, we might use expressions to model supply and demand. Simplifying these expressions can help us understand market trends and make informed decisions.

Moreover, the ability to simplify expressions enhances our problem-solving skills and logical thinking. It teaches us to break down complex problems into smaller, more manageable parts and to apply mathematical rules and principles systematically. These skills are valuable not only in mathematics but also in many other areas of life.

Conclusion

Simplifying expressions by combining like terms is a fundamental skill in mathematics. It involves identifying terms with the same variable and exponent, adding or subtracting their coefficients, and writing the terms in descending order of powers. This process makes expressions easier to understand, manipulate, and use in further calculations. The ability to simplify expressions is essential for success in algebra and other advanced mathematical fields, as well as in various real-world applications. By mastering this skill, we enhance our problem-solving abilities and develop a deeper understanding of mathematical concepts.