(b%2Bc-a)(c%2Ba-b)(a%2Bb-c).jpeg "(a+b+c)(b+c-a)(c+a-b)(a+b-c)")
Factoring and Simplifying (a+b+c)(b+c-a)(c+a-b)(a+b-c)
This expression appears complex, but it can be simplified significantly using basic algebraic techniques. Here's how we can factor and simplify it:
Understanding the Structure
The expression consists of four factors, each of which is a sum or difference of three variables:
- (a + b + c)
- (b + c - a)
- (c + a - b)
- (a + b - c)
Notice that each factor has two positive terms and one negative term. This suggests a pattern we can exploit.
Factoring by Grouping
We can rewrite the expression to make the pattern more evident:
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Rearrange the factors: Group the terms with the same sign together:
(a + b + c)(b + c - a)(c + a - b)(a + b - c) = [(a + b) + c][(b + c) - a][(c + a) - b][(a + b) - c] -
Apply the difference of squares pattern: Recall that (x + y)(x - y) = x² - y². We can apply this pattern to the first two factors and the last two factors:
[(a + b) + c][(b + c) - a][(c + a) - b][(a + b) - c] = [((a + b)² - c²)][((c + a)² - b²)] -
Expand the squares:
[((a + b)² - c²)][((c + a)² - b²)] = [(a² + 2ab + b² - c²)][(c² + 2ac + a² - b²)] -
Rearrange the terms: Group the terms with common variables:
[(a² + 2ab + b² - c²)][(c² + 2ac + a² - b²)] = [(a² + 2ab + b² - c²) + (c² + 2ac + a² - b²)] -
Combine like terms:
[(a² + 2ab + b² - c²) + (c² + 2ac + a² - b²)] = 2a² + 2ab + 2ac + 2b² + 2c² -
Factor out a 2:
2a² + 2ab + 2ac + 2b² + 2c² = 2(a² + ab + ac + b² + c²)
Final Result
Therefore, the simplified form of the expression is: 2(a² + ab + ac + b² + c²). This expression cannot be further factored using real numbers.