Simplifying Algebraic Expressions Combining Like Terms

Introduction

In algebra, simplifying expressions is a fundamental skill. It allows us to rewrite complex expressions in a more manageable form, making them easier to understand and work with. One of the key techniques in simplifying algebraic expressions is combining like terms. This involves identifying terms with the same variable and exponent, and then adding or subtracting their coefficients. This article will guide you through the process of simplifying expressions by combining like terms, with a focus on arranging terms in descending order of their variable exponents.

Understanding Like Terms

Before we dive into the simplification process, it's crucial to understand what like terms are. Like terms are terms that have the same variable raised to the same power. The coefficient, which is the numerical factor in front of the variable, can be different. For example, $3x^2$ and $-5x^2$ are like terms because they both have the variable $x$ raised to the power of 2. However, $3x^2$ and $3x$ are not like terms because the exponents of the variable $x$ are different (2 and 1, respectively). Similarly, $3x^2$ and $3y^2$ are not like terms because they have different variables.

To effectively combine like terms, we need to identify them within an expression. This involves carefully examining each term and comparing its variable and exponent to those of other terms. Once like terms are identified, we can proceed to the next step: adding or subtracting their coefficients.

The Process of Combining Like Terms

The process of combining like terms involves the following steps:

1. Identify like terms: Look for terms that have the same variable raised to the same power.
2. Combine the coefficients: Add or subtract the coefficients of the like terms. Remember to pay attention to the signs (positive or negative) of the coefficients.
3. Write the result: Write the new term with the combined coefficient and the original variable and exponent.

Let's illustrate this process with an example. Consider the expression $5x + 3x - 2x$. All three terms are like terms because they have the same variable, $x$, raised to the same power, 1. To combine them, we add and subtract their coefficients: $5 + 3 - 2 = 6$. Therefore, the simplified expression is $6x$.

Another example is $7y^2 - 4y^2 + y^2$. Again, all terms are like terms. Combining the coefficients, we get $7 - 4 + 1 = 4$. So, the simplified expression is $4y^2$.

Arranging Terms in Descending Order of Power

In addition to combining like terms, it's often helpful to arrange the terms in an expression in descending order of their variable exponents. This means writing the term with the highest power first, followed by the term with the next highest power, and so on, until the constant term (the term without a variable) is written last. This arrangement makes the expression easier to read and understand, and it's a standard convention in algebra.

For example, consider the expression $3x^2 + 5x - 2 + x^3$. To arrange the terms in descending order of power, we first identify the term with the highest power, which is $x^3$. Then, we find the term with the next highest power, which is $3x^2$. Next is $5x$, and finally, the constant term is $-2$. So, the expression arranged in descending order of power is $x^3 + 3x^2 + 5x - 2$.

This arrangement is particularly useful when dealing with polynomials, which are algebraic expressions consisting of one or more terms, each of which is a constant multiplied by a variable raised to a non-negative integer power. Arranging the terms in descending order of power makes it easier to identify the degree of the polynomial, which is the highest power of the variable in the polynomial.

Example: Simplifying and Arranging an Expression

Now, let's apply these concepts to simplify and arrange the expression given in the prompt: $4a^2 - 12a^2 - 3a + 8 + 9a^2 - 1 - 9a + 8a$.

1. Identify Like Terms

First, we identify like terms in the expression. We have three terms with $a^2$: $4a^2$, $-12a^2$, and $9a^2$. We also have three terms with $a$: $-3a$, $-9a$, and $8a$. Finally, we have two constant terms: $8$ and $-1$.

2. Combine Like Terms

Next, we combine like terms. Let's start with the $a^2$ terms: $4a^2 - 12a^2 + 9a^2$. Adding the coefficients, we get $4 - 12 + 9 = 1$. So, the combined term is $1a^2$, which we can simply write as $a^2$.

Now, let's combine the $a$ terms: $-3a - 9a + 8a$. Adding the coefficients, we get $-3 - 9 + 8 = -4$. So, the combined term is $-4a$.

Finally, let's combine the constant terms: $8 - 1 = 7$.

3. Write the Simplified Expression

After combining like terms, our simplified expression is $a^2 - 4a + 7$.

4. Arrange Terms in Descending Order of Power

In this case, the terms are already arranged in descending order of power: $a^2$ (power 2), $-4a$ (power 1), and $7$ (power 0). So, the final simplified expression, arranged in descending order of power, is $a^2 - 4a + 7$.

Additional Tips and Strategies

• Use different shapes or colors to identify like terms: This can be especially helpful when dealing with complex expressions with many terms. For example, you could circle all the $x^2$ terms, underline all the $x$ terms, and box all the constant terms.
• Rewrite the expression to group like terms together: This can make it easier to combine them. For example, in the expression $4a^2 - 12a^2 - 3a + 8 + 9a^2 - 1 - 9a + 8a$, you could rewrite it as $(4a^2 - 12a^2 + 9a^2) + (-3a - 9a + 8a) + (8 - 1)$.
• Be careful with signs: Pay close attention to the signs (positive or negative) of the coefficients when combining like terms. A common mistake is to forget to include the negative sign when subtracting terms.
• Double-check your work: After simplifying an expression, it's always a good idea to double-check your work to make sure you haven't made any mistakes. You can do this by substituting a value for the variable in both the original expression and the simplified expression. If the results are the same, then you've likely simplified the expression correctly.

Conclusion

Combining like terms is a fundamental skill in algebra that allows us to simplify expressions and make them easier to work with. By identifying terms with the same variable and exponent, adding or subtracting their coefficients, and arranging the terms in descending order of power, we can rewrite complex expressions in a more manageable form. This skill is essential for solving equations, graphing functions, and tackling more advanced algebraic concepts. Practice is key to mastering this technique, so work through plenty of examples and don't hesitate to seek help if you get stuck. With consistent effort, you'll become proficient at simplifying algebraic expressions and confidently tackle any mathematical challenge that comes your way.