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 13:
(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 problemsolving skills.