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

Jasonbradley

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Jun 16, 2024

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Expanding and Simplifying the Expression (a+b+c)(a+b-c)(a-b+c)(-a+b+c)

This expression represents the product of four factors, each containing three terms. Let's explore how to expand and simplify it.

Method 1: Direct Expansion

We can expand the expression by repeatedly multiplying two factors at a time.

• Step 1: Multiply the first two factors: (a+b+c)(a+b-c) = a² + 2ab + b² - c²

• Step 2: Multiply the result from Step 1 with the third factor: (a² + 2ab + b² - c²)(a-b+c) = a³ + ab² + ac² - a²b - 2abc - b³ + abc + b²c - c³

• Step 3: Multiply the result from Step 2 with the last factor: (a³ + ab² + ac² - a²b - 2abc - b³ + abc + b²c - c³)(-a+b+c) = -a⁴ - a²b² - a²c² + a³b + 2a²bc + ab³ - abc² - b²c² + c⁴

Simplified Expression:

After combining like terms, the final simplified expression is:

-a⁴ - a²b² - a²c² + a³b + 2a²bc + ab³ - abc² - b²c² + c⁴

Method 2: Using the Difference of Squares

We can simplify the expression by recognizing that the first two and the last two factors are in the form of the difference of squares:

• First two factors: (a+b+c)(a+b-c) = [(a+b) + c][(a+b) - c] = (a+b)² - c²
• Last two factors: (a-b+c)(-a+b+c) = [c + (a-b)][c - (a-b)] = c² - (a-b)²

Now, the expression becomes:

[(a+b)² - c²][c² - (a-b)²]

Applying the difference of squares again:

(a+b)²c² - c⁴ - (a-b)²c² + (a-b)⁴

Expanding and simplifying:

2a²b² + 2a²c² + 2b²c² - a⁴ - b⁴ + c⁴

Conclusion

Both methods lead to the same simplified expression, albeit in different forms. The choice of method depends on personal preference and the specific context of the problem. The simplified expression demonstrates the importance of recognizing patterns in algebraic expressions and utilizing them for simplification.