Problem 20 - Algebra and Trigonometry

Question

\(15-20\) : Use properties of real numbers to write the expression without parentheses. $$ (3 a)(b+c-2 d) $$

Step-by-Step Solution

Answer

The expression simplifies to \(3ab + 3ac - 6ad\).

Step 1: Expand the Expression

Apply the distributive property, which states that \((x)(y+z) = (x)(y) + (x)(z)\), to remove the parentheses. Distribute \(3a\) to each term inside the parentheses: \((3a)(b) + (3a)(c) - (3a)(2d)\).

Step 2: Simplify the Terms

Multiply the terms. So, we multiply each term by \(3a\):\(3ab + 3ac - 6ad\).

Step 3: Final Simplified Expression

Thus, the expression without parentheses becomes \(3ab + 3ac - 6ad\).

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra that helps in simplifying expressions with parentheses. It states that when you multiply a number by a sum or a difference, you can distribute the multiplication across each term inside the parentheses. In mathematical terms, it can be expressed as

• \((x)(y + z) = (x)(y) + (x)(z)\).

This means you multiply \(x\) by each term within the parentheses individually.
For example, if you have an expression like \((3a)(b+c-2d)\), according to the distributive property, you will first multiply \(3a\) with \(b\), then \(3a\) with \(c\), and finally \(3a\) with \(-2d\). This results in the expression \(3ab + 3ac - 6ad\).
Understanding this property is crucial because it allows you to simplify expressions into more manageable terms without parentheses.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations. They are used to represent values and relationships in a concise way. For example:

• Numbers: 5, -2, 3/4
• Variables: \(x\), \(y\), \(z\)
• Operations: addition (+), subtraction (−), multiplication (×), division (÷)

An example of an algebraic expression is \((3a)(b+c-2d)\), where \(a\), \(b\), \(c\), and \(d\) are variables. They represent unknown values and are often used in equations and expressions.
When working with algebraic expressions, it is important to follow the order of operations for accurate solution. In expressions involving parentheses, the operations inside the parentheses are performed first, unless dictated otherwise by the problem.

Expression Simplification

Expression simplification involves reducing an algebraic expression into its most basic, easily understandable form. This often includes distributing, combining like terms, and rewriting expressions to eliminate any unnecessary complexity.
For our exercise \((3a)(b+c-2d)\), simplification involves applying the distributive property: first, we distribute \(3a\) across the terms in the parentheses, resulting in \((3a)(b) + (3a)(c) - (3a)(2d)\), which translates to \(3ab + 3ac - 6ad\).
The goal of simplifying an expression is to present it in a way that's easier to work with or understand, without changing the expression's value. Simplification can make complex algebraic expressions more manageable and is essential for solving problems effectively.