Problem 69 - Algebra 2

Question

PREREQUISITE SKILL. Use the Distributive Property to find each product. $$ 4(x+2) $$

Step-by-Step Solution

Answer

4(x+2) simplifies to 4x + 8.

Step 1: Identify the Expression

The expression given is \(4(x + 2)\). This expression consists of a number outside the parenthesis (4) and a binomial inside the parenthesis (\(x + 2\)).

Step 2: Apply the Distributive Property

The Distributive Property states that \(a(b + c) = ab + ac\). Here, we apply this property by multiplying the number outside the parenthesis (4) with each term inside the parenthesis (\(x + 2\)).

Step 3: Multiply the First Term

Multiply 4 by \(x\): \(4 \times x = 4x\).

Step 4: Multiply the Second Term

Multiply 4 by 2: \(4 \times 2 = 8\).

Step 5: Combine the Products

Combine the results from the previous steps to get the expression \(4x + 8\).

Key Concepts

Expression Simplification

Expression simplification means making an algebraic expression easier to work with by removing any unnecessary complexity. In algebra, this typically involves performing operations, such as addition, subtraction, multiplication, and division, to consolidate and reduce terms.
When we use the distributive property, it's often to simplify expressions by removing parentheses. Once you apply this property, you break down what looks complicated at first into something much simpler. In the current exercise \(4(x + 2)\), by applying the distributive property, the expression simplifies to \(4x + 8\).
Whether dealing with numbers, variables, or both, you simplify an expression so it's as straightforward and understandable as possible. Simplification helps us evaluate the expression quickly, manage complex calculations, and identify equivalent expressions.

Binomial Multiplication

Binomial multiplication involves multiplying a binomial, which is an algebraic expression containing two terms, by another number or expression. It's common to use the distributive property for this purpose.
In the given expression \(4(x + 2)\), the term \(x + 2\) is a binomial. The multiplication here involves multiplying each term of the binomial by the number 4.

• The first multiplication: \(4 imes x = 4x\)
• The second multiplication: \(4 imes 2 = 8\)

These steps separate the operation into manageable pieces. Then, when you combine the results, you have successfully multiplied the binomial by 4 to get the simplified result: \(4x + 8\).
This process harnesses the distributive property to ensure that each term in the binomial is multiplied evenly, which is crucial for accuracy in algebra.

Algebraic Expressions

Algebraic expressions are mathematical statements that include numbers, variables, and operations. They form the building blocks of algebra, used to represent real-world quantities in a compact form that's easy to work with.
The expression given in the exercise \(4(x + 2)\) is an example of an algebraic expression involving a variable \(x\) and constants.

• Such expressions can represent many things, such as equations or functions.
• The goal of working with algebraic expressions is often to simplify or solve them.

By learning to manipulate these expressions, including through techniques like distribution and binomial multiplication, you can solve complex problems in mathematics and understand relationships between variables.
Algebraic expressions are central to learning more advanced math topics. Mastery of this concept requires familiarity with how to handle variables, coefficients, and the operations affecting them.