Problem 3 - Algebra 1: Concepts and Skills

Question

Identify the like terms in the expression. \(2 t+t^{2}+6 t^{2}-6 t\)

Step-by-Step Solution

Answer

The like terms in the expression are \( 2t \) and \( -6t \), and \( t^2 \) and \( 6t^2 \).

Step 1: Identify the Variable Terms

Go through the entire expression and mark the terms containing the variable \( t \) and \( t^2 \). In the given expression \( 2 t+t^{2}+6 t^{2}-6 t \), the terms are \( 2t \), \( -6t \), \( t^2 \) and \( 6t^2 \).

Step 2: Group the Like Terms

Group together the terms which have the same variable and power. Here, the terms \( 2t \) and \( -6t \) have the same variable \( t \) with power 1. And the terms \( t^2 \) and \( 6t^2 \) have the same variable \( t \) with power 2. Thus, \( 2t \) and \( -6t \) are like terms, and \( t^2 \) and \( 6t^2 \) are like terms.

Key Concepts

Variable Terms

In algebra, variable terms are essential components of expressions. Each term consists of a coefficient (a numerical factor) and variables (often represented by letters). The variables indicate the part of the expression that can change or vary. In the expression provided, the variable is \( t \).

Let's consider the expression \( 2t + t^2 + 6t^2 - 6t \). Here, each term possesses a variable \( t \):

• \( 2t \) has a coefficient of 2.
• \( -6t \) has a coefficient of -6.
• \( t^2 \) and \( 6t^2 \) involve \( t \) with exponents.

The presence of a variable in each term allows us to perform operations such as addition and subtraction among terms that are 'like terms.' By identifying variable terms correctly, you can manage and simplify expressions effectively.

Exponents in Algebra

Exponents are powerful mathematical tools that show how many times a number (the base) is multiplied by itself. When looking at variable terms, exponents reveal the degree or power of the variable in each term. For example, in the expression \( t^2 \), the number 2 is an exponent indicating that \( t \) is used as a factor twice: \( t \times t \).

In the context of our exercise, we find two distinct variable terms based on their exponents:

• Terms like \( 2t \) and \( -6t \) have \( t \) with an exponent of 1, often written simply as \( t \).
• Terms like \( t^2 \) and \( 6t^2 \) have \( t \) with an exponent of 2.

Differentiating between these exponents helps in grouping like terms accurately. Remember, only terms with the same variable and exponent are considered like terms and can be combined or rearranged for simplification.

Grouping Like Terms

Grouping like terms is a crucial step in simplifying algebraic expressions. Like terms are those that share the same variables and exponents. This means they have exactly the same variable raised to the same power, allowing them to be added or subtracted from each other.

In the given expression \( 2t + t^2 + 6t^2 - 6t \), we begin by:

• Identifying like terms with \( t \) as the variable, such as \( 2t \) and \( -6t \).
• Identifying like terms with \( t^2 \) as the variable, such as \( t^2 \) and \( 6t^2 \).

Once identified, you grouped these terms. For instance:

• Combine \( 2t \) and \( -6t \) to express \( -4t \).
• Combine \( t^2 \) and \( 6t^2 \) to express \( 7t^2 \).

These grouped terms provide a clearer, simplified expression. Understanding the grouping of like terms is vital for managing complex algebraic equations and improving your solving skills.