Problem 36 - Prealgebra

Question

Apply the distributive property to expression, and then simplify. \(8(3+x)\)

Step-by-Step Solution

Answer

The simplified expression is \(24 + 8x\).

Step 1: Understand the Distributive Property

The distributive property states that for any numbers \( a, b, \) and \( c \), the expression \( a(b + c) \) is equal to \( ab + ac \). This means you multiply the outside number by each of the numbers inside the parentheses, and then add the results.

Step 2: Apply the Distributive Property

Using the distributive property, multiply the number \( 8 \) by each term inside the parentheses in the expression \( 8(3 + x) \). This gives us \((8 \times 3) + (8 \times x)\).

Step 3: Perform the Multiplication

Calculate the products: \(8 \times 3 = 24\) and \(8 \times x = 8x\). Thus, the expression becomes \(24 + 8x\).

Step 4: Combine Like Terms (If Any)

In this case, there are no like terms to combine, so the expression \(24 + 8x\) is already simplified.

Key Concepts

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra. It involves reducing an expression to its most basic form while maintaining its value. The goal is to make the expression easier to work with by eliminating unnecessary parts. For example, consider the expression obtained after applying the distributive property: \(24 + 8x\). There are two terms here: \(24\) and \(8x\).

To simplify an expression, you often:

• Use operations like addition, subtraction, multiplication, and division.
• Combine like terms, which we couldn't do here because there are no like terms in \(24 + 8x\).
• Apply rules like the distributive property to break down or expand parts of the expression.

In our example, since there's nothing more to combine or simplify, \(24 + 8x\) is as simple as it gets. Understanding this process helps you handle more complex equations and functions later in your algebra journey.

Multiplying Polynomials

Multiplying polynomials is an extension of multiplication. It uses similar principles but often involves dealing with multiple terms separately. While our example \(8(3 + x)\) is a straightforward case, understanding polynomial multiplication in larger expressions is crucial.

Here are some key points:

• Each term in one polynomial must be multiplied by every term in the other polynomial.
• Use distributive property for distributing terms across any sums or differences they multiply.
• After multiplying, like terms should be combined to simplify the expression.

Imagine expanding \((a + b)(c + d)\). It requires multiplying \(a\) and \(b\) by both \(c\) and \(d\), resulting in four terms that need to be added together: \(ac + ad + bc + bd\). In our exercise, we used distributive property principles to simplify a single-term expansion, good practice for more complex scenarios.

Algebraic Concepts

Algebra is rich with concepts that help us solve problems involving unknown variables. Understanding algebraic concepts like the distributive property is critical. This property allows you to "distribute" a number over a sum or difference within parentheses, which is crucial when working with expressions and equations.

Here's why it's important:

• It simplifies the multiplication process, especially with variables.
• Helps in rearranging and simplifying equations for easier solving.
• Essential for expanding expressions in polynomial multiplication.

For instance, in \(8(3+x)\), we used the distributive property to expand the expression into \(24 + 8x\), making it ready for further evaluation or use in larger equations. This principle is also foundational for more advanced topics like factoring and solving systems of equations. Mastering these concepts provides a strong base for continuing studies in mathematics.