Simplifying Algebraic Expressions A Step-by-Step Guide

Algebraic expressions are the foundation of mathematics, and mastering the art of simplifying them is crucial for success in higher-level math courses. This comprehensive guide will walk you through the process of simplifying algebraic expressions, providing clear explanations, examples, and step-by-step instructions. We'll focus on combining like terms, using the distributive property, and applying the order of operations to achieve the simplest form of an expression. In this article, we'll specifically address the expression

${ (3x^2 + 5x - 8) + (5x^2 - 13x - 5) }$

and break down the process of simplifying it.

Understanding Algebraic Expressions

Before we dive into the simplification process, let's define what algebraic expressions are and the key components they consist of. 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. Terms are the individual components of an expression separated by addition or subtraction signs. Understanding the anatomy of an algebraic expression is the first step towards simplifying it effectively.

Identifying 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 because they both have the variable ${ x }$ raised to the power of 2. Similarly, ${ 5x }$ and ${ -13x }$ are like terms. However, ${ 3x^2 }$ and ${ 5x }$ are not like terms because they have different powers of ${ x }$. The ability to identify like terms is crucial because only like terms can be combined. In our example expression,

${ (3x^2 + 5x - 8) + (5x^2 - 13x - 5) }$,

the like terms are ${ 3x^2 }$ and ${ 5x^2 }$, ${ 5x }$ and ${ -13x }$, and the constants ${ -8 }$ and ${ -5 }$.

Combining Like Terms

Combining like terms is the core of simplifying algebraic expressions. To combine like terms, you simply add or subtract their coefficients (the numerical part of the term) while keeping the variable and its exponent the same. For instance, to combine ${ 3x^2 }$ and ${ 5x^2 }$, you add their coefficients (3 + 5) to get 8, resulting in ${ 8x^2 }$. Similarly, combining ${ 5x }$ and ${ -13x }$ involves adding their coefficients (5 + (-13) = -8), resulting in ${ -8x }$. Combining the constants ${ -8 }$ and ${ -5 }$ gives ${ -13 }$. Mastering the skill of combining like terms is essential for simplifying any algebraic expression.

Step-by-Step Simplification of the Expression

Now, let's apply these concepts to simplify the given expression:

${ (3x^2 + 5x - 8) + (5x^2 - 13x - 5) }$

Step 1: Remove Parentheses

The first step in simplifying this expression is to remove the parentheses. Since we are adding the two expressions, we can simply rewrite the expression without the parentheses:

${ 3x^2 + 5x - 8 + 5x^2 - 13x - 5 }$

When there is a plus sign between the parentheses, the terms inside the parentheses retain their original signs. If there were a minus sign, we would need to distribute the negative sign to each term inside the second set of parentheses, changing their signs accordingly. However, in this case, we can directly proceed to the next step.

Step 2: Identify Like Terms

Next, we identify the like terms in the expression. As discussed earlier, the like terms are:

• ${ 3x^2 }$ and ${ 5x^2 }$
• ${ 5x }$ and ${ -13x }$
• ${ -8 }$ and ${ -5 }$

It's helpful to rearrange the expression to group like terms together. This can make the combination process clearer and reduce the chance of errors. Rearranging the terms, we get:

${ 3x^2 + 5x^2 + 5x - 13x - 8 - 5 }$

Step 3: Combine Like Terms

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

• Combine ${ 3x^2 }$ and ${ 5x^2 }$: ${ 3x^2 + 5x^2 = 8x^2 }$
• Combine ${ 5x }$ and ${ -13x }$: ${ 5x - 13x = -8x }$
• Combine ${ -8 }$ and ${ -5 }$: ${ -8 - 5 = -13 }$

Step 4: Write the Simplified Expression

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

${ 8x^2 - 8x - 13 }$

Therefore, the simplified form of the expression

${ (3x^2 + 5x - 8) + (5x^2 - 13x - 5) }$

is

${ 8x^2 - 8x - 13 }$.

Analyzing the Answer Choices

Now that we have simplified the expression, let's look at the answer choices provided:

A. ${ 8x^2 + 8x - 13 }$ B. ${ 8x^2 - x - 13 }$ C. ${ 8x^2 - 8x + 13 }$ D. ${ 8x^2 - 8x - 13 }$

Comparing our simplified expression, ${ 8x^2 - 8x - 13 }$, with the answer choices, we can see that option D is the correct answer.

Common Mistakes to Avoid

Simplifying algebraic expressions can be tricky, and it's easy to make mistakes if you're not careful. Here are some common mistakes to avoid:

1. Combining unlike terms: Only like terms can be combined. Make sure you are adding or subtracting terms with the same variable and exponent.
2. Incorrectly distributing the negative sign: When subtracting expressions in parentheses, remember to distribute the negative sign to all terms inside the parentheses. For example,

${ (a + b) - (c + d) = a + b - c - d }$

3. Forgetting to combine all like terms: Double-check your work to ensure you've combined all the like terms in the expression.
4. Making arithmetic errors: Be careful with addition and subtraction, especially when dealing with negative numbers.

Practice Problems

To solidify your understanding of simplifying algebraic expressions, try these practice problems:

1. Simplify: ${ (2x^2 - 3x + 1) + (x^2 + 5x - 4) }$
2. Simplify: ${ (4y^2 + 2y - 7) - (2y^2 - y + 3) }$
3. Simplify: ${ 3(z^2 - 2z + 1) + 2(z^2 + 4z - 3) }$

Working through these problems will help you build confidence and proficiency in simplifying algebraic expressions.

Conclusion

Simplifying algebraic expressions is a fundamental skill in mathematics. By understanding the concepts of like terms, the distributive property, and the order of operations, you can confidently simplify a wide range of expressions. Remember to follow a step-by-step approach, double-check your work, and practice regularly to master this essential skill. In the specific example we addressed,

${ (3x^2 + 5x - 8) + (5x^2 - 13x - 5) }$,

we demonstrated how to remove parentheses, identify and combine like terms, and arrive at the simplified form,

${ 8x^2 - 8x - 13 }$.

With consistent effort and practice, you'll become proficient in simplifying algebraic expressions and excel in your mathematics journey.