(a+b+c)(a-b-c) Formula

4 min read Jun 16, 2024

Understanding the (a + b + c)(a - b - c) Formula

The formula (a + b + c)(a - b - c) represents the product of two binomials, where one binomial is the sum of three terms (a, b, and c) and the other is the difference of the same three terms. This formula can be expanded using the distributive property and algebraic manipulation, resulting in a simpler expression.

Expanding the Formula

Distribute the first term (a) of the first binomial:
```
a(a - b - c) = a² - ab - ac
```
Distribute the second term (b) of the first binomial:
```
b(a - b - c) = ab - b² - bc
```
Distribute the third term (c) of the first binomial:
```
c(a - b - c) = ac - bc - c²
```

Combine the results from steps 1-3:

(a + b + c)(a - b - c) = a² - ab - ac + ab - b² - bc + ac - bc - c²

Simplify the expression by combining like terms:

(a + b + c)(a - b - c) = **a² - b² - c² - 2bc**

Key Takeaways

This formula is a useful shortcut to expand the product of two binomials with specific patterns.
The resulting expression is a difference of squares for the first term (a²) and a sum of squares for the remaining terms (-b² - c² - 2bc).
It's important to note that this formula only applies when the two binomials have the same three terms but with opposite signs in the second binomial.

Examples

Example 1: Find the product of (x + 2y + 3z)(x - 2y - 3z).
- Using the formula: (x + 2y + 3z)(x - 2y - 3z) = x² - (2y)² - (3z)² - 2(2y)(3z) = x² - 4y² - 9z² - 12yz
Example 2: Simplify the expression (5 + 2m + 3n)(5 - 2m - 3n).
- Using the formula: (5 + 2m + 3n)(5 - 2m - 3n) = 5² - (2m)² - (3n)² - 2(2m)(3n) = 25 - 4m² - 9n² - 12mn

Applications

This formula can be applied in various mathematical contexts, including:

Algebraic simplification: It can be used to simplify complex expressions involving binomials.
Factoring: The formula can be used in reverse to factorize expressions into two binomials.
Solving equations: It can be utilized to solve equations involving the product of two binomials.

By understanding and applying the (a + b + c)(a - b - c) formula, you can simplify complex algebraic expressions and enhance your problem-solving skills.