Problem 28 - Algebra and Trigonometry

Question

Use properties of real numbers to write the expression without parentheses. \((3 a)(b+c-2 d)\)

Step-by-Step Solution

Answer

The expression without parentheses is \(3ab + 3ac - 6ad\).

Step 1: Understand the Problem

We need to simplify the expression \[(3a)(b+c-2d)\] by removing the parentheses. This involves using the distributive property to eliminate the parentheses.

Step 2: Apply the Distributive Property

The distributive property states that \[x(a + b + c) = xa + xb + xc\]. Apply this to the expression \[(3a)(b + c - 2d)\].

Step 3: Distribute 3a to Each Term

Multiply each term inside the parentheses by 3a: \[(3a) \cdot b + (3a) \cdot c + (3a) \cdot (-2d)\].

Step 4: Simplify Each Term

Calculate each multiplication: 1. \(3a \cdot b = 3ab\) 2. \(3a \cdot c = 3ac\)3. \(3a \cdot (-2d) = -6ad\)

Step 5: Write the Final Expression

Combine the simplified terms to get the expression without parentheses: \[3ab + 3ac - 6ad\].

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to simplify expressions and solve equations more easily. It states that when you multiply a single term by a sum or difference inside parentheses, you can distribute the multiplication to each term inside the parentheses individually. This property can be represented as \( a(b + c) = ab + ac \). This means you multiply \(a\) by both \(b\) and \(c\).

In our original exercise, we have the expression \((3a)(b+c-2d)\). By applying the distributive property, we take \(3a\) and distribute it to each term inside the parentheses: \(b\), \(c\), and \(-2d\). This step is crucial for breaking down the expression into simpler parts, which then makes it easier to manage and further simplify.

Simplifying Expressions

Simplifying expressions helps in making them easier to understand and work with. It involves performing all possible operations within an expression to reduce it to its simplest form. In algebra, this often includes applying properties like the distributive property, combining like terms, and performing arithmetic operations.

When we simplified \((3a)(b + c - 2d)\), we distributed \(3a\) across each term within the parentheses and then performed the multiplications: \(3a \cdot b = 3ab\), \(3a \cdot c = 3ac\), and \(3a \cdot (-2d) = -6ad\).

After distributing and calculating each term, it's important to combine these results into a final expression with no parentheses: \[3ab + 3ac - 6ad\]. This concise form allows for easier computations in subsequent math operations.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations (like addition and multiplication) that represent a mathematical relationship or quantity. They can be simple, like \(x + 2\), or complex, like our given example: \((3a)(b + c - 2d)\).

Understanding algebraic expressions involves knowing how to manipulate them using various algebraic rules, such as the distributive property, to achieve a simpler or more useful form. In doing so, we often rearrange terms, apply arithmetic operations, and eliminate any parentheses to make the expression easier to work with.

For students, getting acquainted with constructing and simplifying algebraic expressions is vital as it forms the basis of solving algebraic equations and more advanced mathematical problems.