Multiplying Functions

Algebra-1

1. Fundamental Concepts

Definition: Multiplying functions involves multiplying the outputs of two or more functions for each input value.
Like Terms: When multiplying polynomials, like terms are those with identical variable/exponent pairs.
Closure Property: The result of multiplying two functions is always another function.

2. Key Concepts

Basic Rule: $$(f \cdot g)(x) = f(x) \cdot g(x)$$

Degree Preservation: The degree of the resulting polynomial is the sum of the degrees of the original polynomials.

Application: Used to model combined rates in physics and economics.

3. Examples

Example 1 (Basic)

Problem: Multiply $$(2x + 3)(x - 4)$$

Step-by-Step Solution:

Apply distributive property: $$2x \cdot x + 2x \cdot (-4) + 3 \cdot x + 3 \cdot (-4)$$
Simplify: $$2x^2 - 8x + 3x - 12$$
Combine like terms: $$2x^2 - 5x - 12$$

Validation: Substitute \(x=1\) → Original: \(2(1) + 3)(1 - 4) = 5(-3) = -15\); Simplified: \(2(1)^2 - 5(1) - 12 = 2 - 5 - 12 = -15\) ✓

Example 2 (Intermediate)

Problem: Multiply $$(x^2 + 2x + 1)(x - 1)$$

Step-by-Step Solution:

Apply distributive property: $$x^2 \cdot x + x^2 \cdot (-1) + 2x \cdot x + 2x \cdot (-1) + 1 \cdot x + 1 \cdot (-1)$$
Simplify: $$x^3 - x^2 + 2x^2 - 2x + x - 1$$
Combine like terms: $$x^3 + x^2 - x - 1$$

Validation: Substitute \(x=1\) → Original: \((1^2 + 2(1) + 1)(1 - 1) = (1 + 2 + 1)(0) = 0\); Simplified: \(1^3 + 1^2 - 1 - 1 = 0\) ✓

4. Problem-Solving Techniques

Visual Strategy: Use a grid or box method to organize the multiplication of terms.
Error-Proofing: Double-check by substituting values into both the original and simplified expressions.
Concept Reinforcement: Practice with a variety of functions including polynomials, rational functions, and trigonometric functions.