Question

Use the Distributive Property to remove the parentheses. $$ 4(2 x-1) $$

Short answer

8x - 4

Step-by-step solution

Identify the Terms Inside the Parentheses

Look at the expression inside the parentheses, which is \[2x - 1\].

Apply the Distributive Property

To use the distributive property, multiply the term outside the parentheses by each term inside the parentheses. In this case, multiply 4 by both \(2x\) and \(-1\).

Multiply Each Term

\(4 \times 2x = 8x\)\(4 \times -1 = -4\)

Write the Final Expression

Combine the results to write the expression without parentheses: \[8x - 4\].

Key concepts

headline of the respective core concept

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. In this example, you encounter the expression \(4(2x - 1)\), which contains a number (4), a variable (x), and operations (multiplication and subtraction). Understanding algebraic expressions is key to solving equations and performing various mathematical operations.The variable in \(2x\) represents an unknown value that can change. Likewise, the entire expression inside the parentheses represents a specific combination of the variable and constants. When we approach problems involving algebraic expressions, the goal is often to simplify or solve these expressions for a given value of x. Knowing this makes it easier to apply properties like the Distributive Property to simplify and solve.

headline of the respective core concept

Multiplication of terms is a fundamental concept in algebra. Here, you'll see how it’s used in distributive property. When multiplying terms, you need to apply the multiplication operation to each term individually. In the given exercise \(4(2x - 1)\), multiplication of terms involves the following steps:
- **Step 1:** Identify each term in the parentheses. In this example, the terms are \(2x\) and \(-1\).
- **Step 2:** Distribute the factor outside the parentheses (which is 4) to each term inside. This means you will multiply 4 by \(2x\) and 4 by \(-1\).
We have:
\[4 \times 2x = 8x\]
\[4 \times -1 = -4\]
Now you've used the distributive property to multiply each term individually, resulting in new terms that retain the balance of the original equation.

headline of the respective core concept

Simplifying expressions involves combining and reducing terms to make them as simple as possible. After applying the distributive property in the exercise, we obtained the terms \(8x\) and \(-4\). The final step is to write these combined terms as a simpler, more understandable expression:
- **8x - 4**
In this case, there are no like terms to combine further, so the expression \(8x - 4\) is the simplest form. The simplification process makes complex expressions more manageable and paves the way for solving equations efficiently. It’s essential to recognize when an expression is fully simplified to avoid unnecessary steps.Whether you are simplifying after distribution or combining like terms, practice will make this process more intuitive.