(a+b+c)(b+c-a)(c+a-b)(a+b-c) Solution

Suzhoumeite

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Jul 03, 2024

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Solving the Expression: (a+b+c)(b+c-a)(c+a-b)(a+b-c)

In this article, we will explore the solution to the expression (a+b+c)(b+c-a)(c+a-b)(a+b-c). This expression may seem complex at first, but with some algebraic manipulations, we can simplify it to a surprising result.

Step 1: Expanding the Expression

Let's start by expanding each of the factors in the expression:

(a+b+c)(b+c-a) = (a+b+c)(b+c-a) = ab + ac - a^2 + bc + c^2 - b^2

(c+a-b)(a+b-c) = (c+a-b)(a+b-c) = ac + a^2 - bc - c^2 + ab - b^2

Step 2: Multiplying the Expanded Factors

Now, let's multiply the expanded factors together:

(ab + ac - a^2 + bc + c^2 - b^2)(ac + a^2 - bc - c^2 + ab - b^2) = (ab + ac - a^2 + bc + c^2 - b^2)(ac) + (ab + ac - a^2 + bc + c^2 - b^2)(a^2) + ... = (a^2c + abc - a^3c + abc^2 + ac^3 - ab^2c) + (a^3c + a^2bc - a^4c + a^2bc^2 + a^3c^2 - a^3b^2c) + ... = a^4 - b^4 - c^4

The Final Answer

Simplifying the expression further, we get:

(a+b+c)(b+c-a)(c+a-b)(a+b-c) = a^4 - b^4 - c^4

This result is quite interesting, as it shows that the original complex expression can be reduced to a simple difference of fourth powers!

In conclusion, we have successfully solved the expression (a+b+c)(b+c-a)(c+a-b)(a+b-c) and arrived at a surprising and elegant solution.