(b%2Bc-a)(c%2Ba-b)(a%2Bb-c)-by-formula.jpeg "(a+b+c)(b+c-a)(c+a-b)(a+b-c) By Formula")
The Fabulous Formula: (a+b+c)(b+c-a)(c+a-b)(a+b-c)
Have you ever come across a mathematical expression that seems daunting at first, but suddenly becomes manageable once you apply the right formula? Today, we're going to explore one such formula that has fascinated mathematicians for centuries: (a+b+c)(b+c-a)(c+a-b)(a+b-c). Buckle up, because we're about to embark on a thrilling adventure of algebraic manipulations!
The Formula: A Brief Introduction
The formula in question is a product of four binomials, each containing variables a, b, and c. At first glance, it may seem like a complex expression, but fear not! With the right techniques and a dash of creativity, we can simplify this formula to reveal its underlying beauty.
Step 1: Expanding the Formula
To begin, let's expand each of the four binomials:
(a+b+c) = a + b + c
(b+c-a) = b + c - a
(c+a-b) = c + a - b
(a+b-c) = a + b - c
Now, let's multiply these four expressions together:
(a + b + c)(b + c - a)(c + a - b)(a + b - c)
Step 2: Simplifying the Expression
As we multiply the four binomials, we'll start to notice some fascinating patterns. Let's focus on the first two binomials:
(a + b + c)(b + c - a) = (a + b + c)(b + c) - (a + b + c)a
Expanding the right-hand side, we get:
= ab + ac + bc + bc + cc - a^2 - ab - ac
Simplify the expression by combining like terms:
= bc + cc - a^2
Next, let's multiply the result by the third binomial:
(bc + cc - a^2)(c + a - b)
Expanding and simplifying, we get:
= bc^2 + cc^2 - a^2c - ab^2 - a^2b + abc
Now, it's time to bring in the fourth binomial:
(bc^2 + cc^2 - a^2c - ab^2 - a^2b + abc)(a + b - c)
After another round of expansions and simplifications, we're left with:
= a^2b^2 + a^2c^2 + b^2c^2 - a^4 - b^4 - c^4
The Final Result
And there you have it! The formula (a+b+c)(b+c-a)(c+a-b)(a+b-c) simplifies to:
a^2b^2 + a^2c^2 + b^2c^2 - a^4 - b^4 - c^4
This formula is a testament to the beauty of algebraic manipulations. By repeatedly applying the distributive property and combining like terms, we've transformed a complex expression into a stunningly simple and symmetric result.
Conclusion
In conclusion, the formula (a+b+c)(b+c-a)(c+a-b)(a+b-c) is a masterclass in algebraic simplification. By breaking down the expression into manageable parts and applying the right techniques, we've uncovered a hidden gem of mathematics. So, the next time you encounter a daunting formula, remember: with persistence and creativity, even the most complex expressions can be tamed!
