91Ó°ÊÓ

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operation symbols. They are used to represent quantities and their relationships in equations. For example, in the expression \(5(t^2 + 6t) - 8\), we have:

• A coefficient (5)
• Variables (\t \ and \( t \))
• Constants (8)

Understanding algebraic expressions allows you to manipulate and solve them effectively. This skill is important for tackling more complex math problems.
Let’s break down what each part of the expression means:

• 5 is the coefficient that multiplies the terms inside the parentheses.
• \t \ and \( t^2 \) represent variables.
• \( -8 \) is a constant that will be subtracted after distributing the 5.

Knowing these elements will help you understand how to multiply and simplify algebraic expressions.

The distribution property, also known as the distributive property, is a useful algebraic rule. It lets you multiply a single term by each term inside a set of parentheses. This will often simplify the equation.
The property states: \(a(b + c) = ab + ac\).
In our given expression \(5(t^2 + 6t) - 8\), we can apply the distributive property to solve it.
Using the Distributive Property:

• Multiply 5 by \( t^2 \): 5 \(t^2 \)= \( 5t^2 \)
• Multiply 5 by 6t: \(5(6t) = 30t\)

Now, rewrite the expression using these products: \(5(t^2 + 6t) = 5t^2 + 30t\). This step helps to simplify the equation by removing the parentheses.
Applying the distributive property correctly ensures that you multiply each term properly, setting a strong foundation for solving the rest of the problem.

Simplification is the process of reducing an expression to its simplest form.
Here, our simplified expression is \( 5t^2 + 30t - 8\). To check its simplification:

• Combine like terms: In our case, there are no like terms left to combine.
• Reduce any terms if possible: There are no further reductions since all terms are distinct.

Therefore, \( 5t^2 + 30t - 8 \) is already simplified.
Key steps in simplification include:

• Combining like terms that have the same variable and exponent.
• Performing arithmetic operations (addition, subtraction, multiplication, and division).

Simplifying helps in making expressions more manageable and easier to understand. This foundation can be built upon for more complicated algebra problems.