Problem 24 - Precalculus Mathematics for Calculus

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 Expression

The given expression is \[(3a)(b+c-2d).\]This is a multiplication problem where we multiply a monomial, \(3a\), with the binomial, \(b+c-2d\). Our goal is to remove the parentheses by applying the distributive property.

Step 2: Apply the Distributive Property

According to the distributive property: \[x(y+z) = xy + xz.\]We will apply this property to the expression \((3a)(b+c-2d)\). This means we will distribute \(3a\) across each term inside the parentheses.

Step 3: Distribute the Monomial

Multiply \(3a\) by each term inside the parentheses:- Multiply \(3a\) by \(b\), resulting in \(3ab\).- Multiply \(3a\) by \(c\), resulting in \(3ac\).- Multiply \(3a\) by \(-2d\), resulting in \(-6ad\).

Step 4: Rewrite Without Parentheses

After distributing, combine the results to write the expression without parentheses:\[3ab + 3ac - 6ad.\] This is the expanded form of the original expression without parentheses.

Key Concepts

Properties of Real Numbers

The properties of real numbers provide a foundation for understanding arithmetic and algebraic operations. These properties include the commutative, associative, identity, inverse, and distributive properties. Each of these properties helps us manipulate and simplify mathematical expressions in different ways.

In this exercise, we focus on the distributive property. It is one of the most useful properties when dealing with expressions involving parentheses. The distributive property allows us to multiply a single term by a sum or difference inside parentheses. It states that for any real numbers, the expression \(a(b + c)\) is equivalent to \(ab + ac\). This helps us "distribute" the term outside the bracket to each term within the bracket efficiently.

Understanding and applying these properties is crucial in expression manipulation and simplification, leading to easier calculations and more organized solutions.

Expression Simplification

Expression simplification is an essential skill in algebra where we try to reduce an expression to its simplest form. This often involves eliminating parentheses, combining like terms, and following the order of operations.

In our example, \((3a)(b+c-2d)\), we aim to simplify the expression by removing parentheses using the distributive property. This involves several steps, like distributing \(3a\) to each term inside the parentheses and rewriting the expression in its expanded form.

• Simplifying expressions makes them easier to work with and often reveals relationships between different terms that might not be obvious at first.
• It's important to double-check each step to ensure accuracy, especially when signs change, like when distributing over subtraction.

Simplifying expressions also helps in solving equations and inequalities more efficiently, as it reduces the number of terms and calculations involved.

Binomial Multiplication

Binomial multiplication is a key concept when dealing with expressions where two terms are being multiplied, often seen within parentheses. These expressions require careful application of the distributive property to break them down.

For the expression \((3a)(b+c-2d)\), binomial multiplication involves multiplying a single term, \(3a\), by each of the three terms in the binomial \(b+c-2d\). This type of multiplication systematically separates each term in the binomial and individually multiplies it by the monomial.

• This process produces terms like \(3ab\), \(3ac\), and \(-6ad\) once the multiplication is done.
• Each product is a part of the final simplified expression, helping us expand and see the expression without its initial parentheses.

Understanding binomial multiplication is vital as it builds the foundation for more complex algebraic operations. Once mastered, it can simplify expanding polynomials and solving various algebraic problems.