Expanding Parentheses: The Distributive Law

Article Contents

The Distributive Law
One Term Outside Parentheses
Multiplying Binomials (FOIL)
Multiple Terms Outside
Common Mistakes
Practice Examples
Interactive Exercises

1. The Distributive Law {#the-distributive-law}

The distributive law states that multiplication can be "distributed" over addition or subtraction:

a (b + c) = ab + a c

a (b - c) = ab - a c

Visual Explanation

You have $3$ groups of $(4 + 2)$ :

3 \cdot (4 + 2) = 3 \cdot 4 + 3 \cdot 2 = 12 + 6 = 18

2. One Term Outside Parentheses {#one-term-outside}

Basic Example

3 (x + 4) = 3 \cdot x + 3 \cdot 4 = 3 x + 12

Negative Sign

- 2 (x - 5) = - 2 \cdot x + (- 2) \cdot (- 5) = - 2 x + 10

Decimal Numbers

0, 5 (2 x + 6) = 1 x + 3 = x + 3

3. Multiplying Binomials (FOIL) {#multiplying-binomials}

When multiplying two binomials, use the FOIL method:

Letter	Meaning	Example $(x + 3) (x + 2)$
First	First terms	$x \cdot x = x^{2}$
Outer	Outer terms	$x \cdot 2 = 2 x$
Inner	Inner terms	$3 \cdot x = 3 x$
Last	Last terms	$3 \cdot 2 = 6$

Result

(x + 3) (x + 2) = x^{2} + 2 x + 3 x + 6 = x^{2} + 5 x + 6

4. Multiple Terms Outside {#multiple-terms-outside}

Two Terms

(2 x + 3) (x - 4)

$2 x \cdot x = 2 x^{2}$
$2 x \cdot (- 4) = - 8 x$
$3 \cdot x = 3 x$
$3 \cdot (- 4) = - 12$

= 2 x^{2} - 8 x + 3 x - 12 = 2 x^{2} - 5 x - 12

5. Common Mistakes {#common-mistakes}

❌ Mistake 1: Forgetting the Sign

- (x + 3) = - x - 3

The minus sign in front of parentheses changes ALL signs inside!

❌ Mistake 2: Incorrect Expansion

3 (x + 4) = 3 x + 4 (WRONG!)

3 (x + 4) = 3 x + 12 (CORRECT)

6. Practice Examples {#practice-examples}

Example 1: Simple Expansion

4 (2 x + 3) = 8 x + 12

Example 2: Two Parentheses

(x + 5) (x - 2) = x^{2} - 2 x + 5 x - 10 = x^{2} + 3 x - 10

Example 3: With Zero

(x + 3) (x - 3) = x^{2} - 9

This is the difference of squares formula!

7. Interactive Exercises {#interactive-exercises}

Practice expanding:

Formula Summary

Formula	Form
One term	$a (b + c) = ab + a c$
FOIL	$(a + b) (c + d) = a c + a d + b c + b d$
Difference of squares	$(a + b) (a - b) = a^{2} - b^{2}$
Perfect square	$(a + b)^{2} = a^{2} + 2 ab + b^{2}$

Introduction to Algebra – basic concepts
Factoring – reverse process
Difference of Squares – useful formula

Expanding Parentheses