Problem 85 - Algebra A Combined Function

Question

Write each algebraic expression described. In your own words, explain how to combine like terms.

Step-by-Step Solution

Answer

Combine like terms by adding or subtracting their coefficients.

Step 1: Understanding Like Terms

Like terms in an algebraic expression are terms that have the same variables raised to the same power. For example, in the expression \(3x^2 + 5x - 2x^2 + 4\), the terms \(3x^2\) and \(-2x^2\) are like terms because they both have the variable \(x^2\). The coefficients (3 and -2 in this case) are added or subtracted.

Step 2: Identify Like Terms

Scan through the given expression to identify terms that have the same variable factors. In the expression \(3x^2 + 5x - 2x^2 + 4\), group the terms \(3x^2\) and \(-2x^2\) together, then note that \(5x\) does not have a like partner, and \(4\) is a constant term.

Step 3: Combine Coefficients of Like Terms

Add or subtract the coefficients of the like terms to combine them into a single term. For \(3x^2\) and \(-2x^2\), compute: \(3 - 2 = 1\), so the terms combined become \(x^2\).

Step 4: Write the Simplified Expression

After combining like terms, write out the simplified expression: \(x^2 + 5x + 4\). No terms in this expression can be further combined, as each term is distinct in its variable factor.

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that are structured similarly to numerical expressions but with unknown components. They can include constants and variables, and operations like addition, subtraction, multiplication, and division. Think of algebraic expressions like sentences that convey mathematical information.
For example, the expression \(3x + 2\) includes a term \(3x\), which can change when the variable \(x\) changes. Meanwhile, the number 2 is a constant term that stays the same.

• **Terms** in an algebraic expression are distinct parts separated by plus or minus signs.
• **Coefficients** are the numerical parts that are multiplied by the variables.
• **Variables** are symbols that represent unknown numbers, often shown as letters like \(x, y, z\).

Understanding algebraic expressions is foundational for solving equations and manipulating expressions to simplify or solve them.

Identifying Like Terms

Identifying like terms is crucial when working with algebraic expressions to make simplification easier. Like terms are terms in an expression that have the same variable raised to the same power.
For example, in the expression \(4x^2 + 3x - x^2\), the like terms \(4x^2\) and \(-x^2\) both include the variable \(x^2\). Recognizing these helps streamline the process.

• Scan your expression for terms with the exact same variable and exponent.
• Constant terms, like standalone numbers, are considered like terms with other constants.
• Group like terms together by their shared components for clarity.

By pinpointing like terms, you can effectively combine them, reducing the complexity of the equation you're working with.

Simplifying Expressions

Simplifying expressions involves reorganizing them to make them less complicated while essentially maintaining their original value. It is a process often applied before solving equations.
To simplify an expression like \(3x^2 + 5x - 2x^2 + 4\), follow these steps:

• Identify and separate like terms within the expression.
• Combine the coefficients of these like terms by performing addition or subtraction. For instance, \(3x^2 - 2x^2\) simplifies to \(x^2\).
• Rewrite the expression with the new combined terms, e.g., \(x^2 + 5x + 4\).

Simplified expressions are easier to interpret, solve, and further manipulate when you're dealing with more complex mathematical problems. This clarity is invaluable in problem-solving scenarios.