Expanding and Collecting Like Terms

Oct 13, 2025 by windailys.com 12 views

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Introduction

Simplifying complex algebraic expressions is a crucial skill in mathematics, particularly in algebra and calculus. One effective method for simplifying such expressions is by expanding and collecting like terms. This technique involves breaking down the expression into its individual components, expanding each term, and then combining like terms to simplify the expression. In this article, we will explore the concept of expanding and collecting like terms, and provide a step-by-step guide on how to apply this technique to simplify complex algebraic expressions.

What are Like Terms?

Like terms are algebraic expressions that have the same variable(s) raised to the same power. In other words, like terms are expressions that have the same combination of variables and exponents. For example, 2x and 4x are like terms because they both have the variable x raised to the power of 1. Similarly, 3x^2 and 5x^2 are like terms because they both have the variable x raised to the power of 2.

Expanding and Collecting Like Terms

Expanding and collecting like terms involves the following steps:

Break down the expression: Break down the given expression into its individual components.
Expand each term: Expand each term in the expression by multiplying out any parentheses or brackets.
Combine like terms: Combine like terms by adding or subtracting the coefficients of the like terms.
Simplify the expression: Simplify the expression by combining any remaining like terms.

Example 1: Simplifying a Complex Expression

Let's consider the following complex expression:

{ \begin{array}{l} 2(4+a) + 10a(6+2b+7(2a-ab)) + (3a^4+11a) + 3a(5a^5-2a^2) \end{array} \}

To simplify this expression, we will follow the steps outlined above.

Step 1: Break down the expression

The given expression can be broken down into the following individual components:

$2(4+a)$
$10a(6+2b+7(2a-ab))$
$(3a^4+11a)$
$3a(5a^5-2a^2)$

Step 2: Expand each term

We will now expand each term in the expression:

$2(4+a) = 8 + 2a$
$10a(6+2b+7(2a-ab)) = 10a(6+2b+14a-7ab) = 60a + 20ab + 140a^2 - 70a^2b$
$(3a^4+11a) = 3a^4 + 11a$
$3a(5a^5-2a^2) = 15a^6 - 6a^3$

Step 3: Combine like terms

We will now combine like terms in the expression:

$8 + 2a + 60a + 20ab + 140a^2 - 70a^2b + 3a^4 + 11a + 15a^6 - 6a^3$
Combine like terms: $8 + (2a + 60a) + 20ab + (140a^2 - 70a^2b) + 3a^4 + 11a + 15a^6 - 6a^3$
Simplify: $8 + 62a + 20ab + 70a^2b + 3a^4 + 11a + 15a^6 - 6a^3$

Step 4: Simplify the expression

We will now simplify the expression by combining any remaining like terms:

$8 + 62a + 20ab + 70a^2b + 3a^4 + 11a + 15a^6 - 6a^3$
Combine like terms: $8 + (62a + 11a) + 20ab + 70a^2b + 3a^4 + 15a^6 - 6a^3$
Simplify: $8 + 73a + 20ab + 70a^2b + 3a^4 + 15a^6 - 6a^3$

The final simplified expression is:

$8 + 73a + 20ab + 70a^2b + 3a^4 + 15a^6 - 6a^3$

Conclusion

Simplifying complex algebraic expressions is a crucial skill in mathematics, particularly in algebra and calculus. By expanding and collecting like terms, we can simplify complex expressions and make them easier to work with. In this article, we have explored the concept of expanding and collecting like terms, and provided a step-by-step guide on how to apply this technique to simplify complex algebraic expressions. By following these steps, you can simplify even the most complex expressions and make them easier to work with.

Example 2: Simplifying a Quadratic Expression

Let's consider the following quadratic expression:

{ \begin{array}{l} x^2 + 4x + 4 \end{array} \}

To simplify this expression, we will follow the steps outlined above.

Step 1: Break down the expression

The given expression can be broken down into the following individual components:

$x^2$
$4x$
$4$

Step 2: Expand each term

We will now expand each term in the expression:

$x^2 = x^2$
$4x = 4x$
$4 = 4$

Step 3: Combine like terms

We will now combine like terms in the expression:

$x^2 + 4x + 4$
Combine like terms: $x^2 + 4x + 4$
Simplify: $x^2 + 4x + 4$

The final simplified expression is:

$x^2 + 4x + 4$

Example 3: Simplifying a Polynomial Expression

Let's consider the following polynomial expression:

{ \begin{array}{l} x^3 + 2x^2 + 3x + 4 \end{array} \}

To simplify this expression, we will follow the steps outlined above.

Step 1: Break down the expression

The given expression can be broken down into the following individual components:

$x^3$
$2x^2$
$3x$
$4$

Step 2: Expand each term

We will now expand each term in the expression:

$x^3 = x^3$
$2x^2 = 2x^2$
$3x = 3x$
$4 = 4$

Step 3: Combine like terms

We will now combine like terms in the expression:

$x^3 + 2x^2 + 3x + 4$
Combine like terms: $x^3 + 2x^2 + 3x + 4$
Simplify: $x^3 + 2x^2 + 3x + 4$

The final simplified expression is:

$x^3 + 2x^2 + 3x + 4$

Conclusion

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Q1: What is the purpose of simplifying algebraic expressions?

A1: The purpose of simplifying algebraic expressions is to make them easier to work with and understand. Simplifying expressions can help to identify patterns and relationships between variables, which can be useful in solving equations and inequalities.

Q2: What are like terms in algebra?

A2: Like terms are algebraic expressions that have the same variable(s) raised to the same power. For example, 2x and 4x are like terms because they both have the variable x raised to the power of 1.

Q3: How do I identify like terms in an expression?

A3: To identify like terms in an expression, look for terms that have the same variable(s) raised to the same power. You can also use the distributive property to expand expressions and make it easier to identify like terms.

Q4: What is the distributive property in algebra?

A4: The distributive property is a rule in algebra that states that a single term can be distributed to multiple terms inside parentheses. For example, 2(x + 3) = 2x + 6.

Q5: How do I expand an expression using the distributive property?

A5: To expand an expression using the distributive property, multiply each term inside the parentheses by the term outside the parentheses. For example, 2(x + 3) = 2x + 6.

Q6: What is the difference between expanding and simplifying an expression?

A6: Expanding an expression involves multiplying out the terms inside parentheses, while simplifying an expression involves combining like terms to make the expression easier to work with.

Q7: How do I simplify an expression by collecting like terms?

A7: To simplify an expression by collecting like terms, combine the coefficients of like terms. For example, 2x + 4x = 6x.

Q8: What is the order of operations in algebra?

A8: The order of operations in algebra is:

Parentheses: Evaluate expressions inside parentheses first.
Exponents: Evaluate any exponential expressions next.
Multiplication and Division: Evaluate any multiplication and division operations from left to right.
Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q9: How do I apply the order of operations to simplify an expression?

A9: To apply the order of operations to simplify an expression, follow the order of operations and evaluate each term in the expression. For example, 2(x + 3) + 4 = 2x + 6 + 4 = 2x + 10.

Q10: What are some common mistakes to avoid when simplifying expressions?

A10: Some common mistakes to avoid when simplifying expressions include:

Forgetting to distribute terms inside parentheses
Not combining like terms
Not following the order of operations
Making errors when multiplying or dividing terms

Q11: How do I check my work when simplifying an expression?

A11: To check your work when simplifying an expression, plug the simplified expression back into the original equation and verify that it is true. You can also use a calculator or computer software to check your work.

Q12: What are some real-world applications of simplifying expressions?

A12: Simplifying expressions has many real-world applications, including:

Solving equations and inequalities in physics and engineering
Modeling population growth and decay in biology
Analyzing financial data in economics
Solving optimization problems in computer science

Q13: How can I practice simplifying expressions?

A13: You can practice simplifying expressions by working through algebraic exercises and problems. You can also use online resources and software to practice simplifying expressions.

Q14: What are some common algebraic expressions that can be simplified?

A14: Some common algebraic expressions that can be simplified include:

Quadratic expressions: x^2 + 4x + 4
Polynomial expressions: x^3 + 2x^2 + 3x + 4
Rational expressions: (x + 2) / (x - 1)

Q15: How can I apply simplifying expressions to real-world problems?

A15: To apply simplifying expressions to real-world problems, identify the variables and constants in the problem and simplify the expression accordingly. For example, if you are modeling population growth and decay, you may need to simplify expressions involving exponential functions.

Conclusion

Simplifying expressions is a crucial skill in algebra and mathematics. By understanding the concepts of like terms, the distributive property, and the order of operations, you can simplify complex expressions and make them easier to work with. By practicing simplifying expressions and applying them to real-world problems, you can develop a deeper understanding of algebra and mathematics.