Problem 17 - Algebra 1

Question

Use the distributive property and mental math to simplify the expression. $$ 8 t^{2}-2 t+5 t-4 $$

Step-by-Step Solution

Answer

The simplified form of the algebraic expression is \(8 t^{2} + 3t - 4\).

Step 1: Identify Like Terms

In this given expression \(8 t^{2}-2 t+5 t-4\), the like terms are \(-2t\) and \(+5t\) as they both have the variable \(t\) to the power of 1.

Step 2: Combine Like Terms

Now, combine these like terms by adding or subtracting their coefficients. Here, \(-2t + 5t\) becomes \(3t\). This will give us the new expression: \(8 t^{2} + 3t - 4\).

Step 3: Final simplified expression

So after combining like terms, the simplified algebraic expression is \(8 t^{2} + 3t - 4\).

Key Concepts

Combining Like Terms

Combining like terms is a fundamental skill in algebra that helps simplify expressions by merging terms that have the same variable and exponent. In the expression \(8t^{2} - 2t + 5t - 4\), like terms are specifically those that share the same variables to the same power.

• Identify like terms: Look for terms in the expression that have identical variable parts. In this example, \(-2t\) and \(+5t\) are like terms because they both involve the variable \(t\) raised to the same power of 1.
• Combine them: Add or subtract the coefficients (numerical parts) of these terms. So, \(-2t + 5t\) can be simplified by calculating \(-2 + 5\), resulting in \(+3t\).

This process reduces the complexity of expressions and is essential for solving equations efficiently. It makes working with algebraic expressions more manageable and sets the stage for further simplification.

Simplifying Expressions

Simplifying expressions involves reducing them to their most compact form while preserving equivalency. This means getting rid of unnecessary complexity without changing the value of the expression
.To simplify \(8t^{2} - 2t + 5t - 4\), we start by combining like terms, which changes the expression to \(8t^{2} + 3t - 4\). This is a simpler version because it has fewer terms.

• Clarity: A simplified expression is easier to understand and work with, and it helps reveal the underlying relationships between variables.
• Efficiency: Simplified expressions are easier to use in further calculations, such as solving equations or finding function values.

Though it may seem like a small change, simplifying expressions is crucial for effective mathematical problem-solving and lays the groundwork for more advanced algebraic concepts.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They are the foundation of algebra and are used to represent mathematical ideas and solve equations. In the expression \(8t^{2} - 2t + 5t - 4\), you see typical components of an algebraic expression
:

• Constants: Fixed values without variables, such as \(-4\).
• Variables: Symbols that represent numbers we don't know yet, like \(t\) in this expression.
• Coefficients: Numbers that multiply the variables, like \(-2\) and \(+5\) which are coefficients of \(t\).
• Exponents: Indicate how many times the base (a variable in this case) is used as a factor, as seen in \(t^{2}\).

Understanding how to manipulate algebraic expressions is essential in algebra. They define equations and functions, and are key in analyzing relationships between changing quantities. Mastering this concept can greatly enhance problem-solving skills and mathematical intuition.