**Algebraic Expression Multiplication: A Comprehensive Tutorial**

Algebra, often perceived as a mathematical puzzle, unveils its depth and intricacy when it comes to multiplying algebraic expressions. In this tutorial, we embark on a journey through the fundamental principles and techniques that govern the multiplication of algebraic expressions, unlocking the door to advanced mathematical reasoning.

**Understanding Algebraic Expressions: A Recap**

Before delving into multiplication, let’s briefly recap what algebraic expressions are. An algebraic expression is a mathematical phrase involving numbers, variables, and operations. Variables, often denoted by letters like $x$, $y$, or $z$, represent unknown or changing quantities, while constants are fixed numerical values. These expressions can be combined, manipulated, and, crucially, multiplied to explore relationships and solve problems.

Table of Contents

## **1. **Multiplication Basics: The Foundation**

**Like Terms in Multiplication:**

The key to mastering multiplication lies in recognizing like terms. Like terms have the same variable and exponent. When multiplying expressions, focus on combining these similar components.

**Example: Multiplying Like Terms – Variables:**

2x . 3x = 6x^{2}

**Example: Multiplying Like Terms – Constants:**

4a . 2a = 8a^{2}

### 2. **Distributive Property: A Powerful Tool**

**Expanding Expressions:**

The distributive property becomes invaluable when multiplying expressions. It allows us to distribute a value to all terms within parentheses.

### 3. **Multiplying Binomials: The FOIL Method**

**FOIL Method Defined:**

Multiplying binomials (expressions with two terms) involves the FOIL method, which stands for First, Outer, Inner, Last. It’s a systematic approach to ensure all terms are multiplied.

**Example: FOIL Method – Binomials:**

### 4. **Multiplying Polynomials: A Gradual Expansion**

**Polynomials Defined:**

Polynomials, expressions with multiple terms, require a gradual approach. Multiply each term in one polynomial by each term in the other, combining like terms as needed.

**Example: Multiplying Polynomials:**

**Solved Examples**

**Illustration 1: Multiply 6x with 10y and 15z**

**Solution**: 6x × 10y × 15z = 60xy × 15z = 900xyz

We multiply the first two monomials and then the resulting monomial to the third monomial.

**Illustration 2:** **Find the volume of a cuboid whose length is 4ax, breadth is 2by and height is 20cz.**

**Solution**:

Volume = length × breadth × height

Therefore, volume = 4ax × 2by × 20cz = 4 × 2 × 20 × (ax) × (by) × (cz) = 160axbycz

**Illustration 3: Multiply (3a ^{2} + 6a + 5) by 4a.**

Solution:

4a × (3a^{2} + 6a + 5)

= (4a × 3a^{2}) + (4a × 6a) + (4a × 5)

= 12a^{3} + 24a^{2} + 20a