Problem 25 - Elementary and Intermediate Algebra

Question

Multiply. See Example 2. $$ 3 x(x+4) $$

Step-by-Step Solution

Answer

The product is \(3x^2 + 12x\).

Step 1: Distribute the First Term

Each term inside the parentheses is multiplied by the term outside. First, multiply the first term:\[3x \times x = 3x^2\].

Step 2: Distribute to the Second Term

Now multiply the second term inside the parentheses by the term outside:\[3x \times 4 = 12x\].

Step 3: Combine the Resulting Terms

The products from Step 1 and Step 2 are combined to form the expression:\[3x(x + 4) = 3x^2 + 12x\].

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra that allows us to multiply a single term by a sum or difference inside parentheses. This property helps us break down expressions into simpler parts and is especially useful when dealing with polynomials. In mathematical terms, the distributive property can be expressed as:

• \( a(b + c) = ab + ac \)
• \( a(b - c) = ab - ac \)

This property ensures that multiplication affects each term within the parentheses individually. In our exercise, we apply the distributive property by multiplying \(3x\) with both \(x\) and \(4\), leading to the terms \(3x^2\) and \(12x\). Using this property ensures that calculations are methodical and correct.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operation symbols. They represent mathematical phrases that can encompass constants (known numbers), variables (unknowns), and operations (like addition and multiplication). For example, in the expression \(3x(x + 4)\), \(3x\) is a monomial, while \((x+4)\) is a binomial.

When working with algebraic expressions, each component has its role. Coefficients (like the 3 in \(3x\)) are numerical factors that multiply the variables. The variables represent unknowns that can change,

• allowing for dynamic relationships within the expression.

By learning how to handle these expressions, we gain the ability to solve a wide spectrum of mathematical problems, from simple equations to complex real-world situations.

Simplifying Expressions

Simplifying expressions involves rewriting them in a more concise way without changing their value. The goal is to make calculations easier and clearer. In this context, we transformed \(3x(x + 4)\) into \(3x^2 + 12x\) by applying the distributive property. Simplifying expressions often entails:

• Combining like terms
• Using arithmetic operations
• Applying basic algebraic principles

When terms are similar (such as both having the same variable raised to the same power), they can be combined, making the expression more streamlined. Simplifying helps in solving equations more efficiently, as it usually reduces the complexity of the problems, leading to quicker solutions.