(a-b)(a-c)(b-c)(a+b)(a+c)(b+c)+(a+b)(b+c)(a+c)(aac+bba+ccb)+abc(a-b)(a-c)(b-c)

asked by guest
on Oct 26, 2024 at 4:31 am



You asked:

Evaluate the expression: \(\left(a - b\right) \cdot \left(a - c\right) \cdot \left(b - c\right) \cdot \left(a + b\right) \cdot \left(a + c\right) \cdot \left(b + c\right) + \left(a + b\right) \cdot \left(b + c\right) \cdot \left(a + c\right) \cdot \left(a a c + b b a + c c b\right) + a b \cdot c \cdot \left(a - b\right) \cdot \left(a - c\right) \cdot \left(b - c\right)\)

MathBot Answer:

Evaluated



\(\displaystyle \left(a - b\right) \cdot \left(a - c\right) \cdot \left(b - c\right) \cdot \left(a + b\right) \cdot \left(a + c\right) \cdot \left(b + c\right) + \left(a + b\right) \cdot \left(b + c\right) \cdot \left(a + c\right) \cdot \left(a a c + b b a + c c b\right) + a b \cdot c \cdot \left(a - b\right) \cdot \left(a - c\right) \cdot \left(b - c\right) = a b c \left(a - b\right) \left(a - c\right) \left(b - c\right) + \left(a - b\right) \left(a - c\right) \left(a + b\right) \left(a + c\right) \left(b - c\right) \left(b + c\right) + \left(a + b\right) \left(a + c\right) \left(b + c\right) \left(a^{2} c + b^{2} a + c^{2} b\right) \)

Expanded

\[\left(a - b\right) \cdot \left(a - c\right) \cdot \left(b - c\right) \cdot \left(a + b\right) \cdot \left(a + c\right) \cdot \left(b + c\right) + \left(a + b\right) \cdot \left(b + c\right) \cdot \left(a + c\right) \cdot \left(a a c + b b a + c c b\right) + a b \cdot c \cdot \left(a - b\right) \cdot \left(a - c\right) \cdot \left(b - c\right) = a^{4} b^{2} + a^{4} b c + a^{3} b^{3} + 3 a^{3} b^{2} c + a^{3} c^{2} b + a^{3} c^{3} + a^{2} b^{3} c + 3 a^{2} b^{2} c^{2} + 3 a^{2} c^{3} b + a^{2} c^{4} + b^{4} a c + 3 b^{3} c^{2} a + b^{2} c^{3} a + c^{4} a b + b^{4} c^{2} + b^{3} c^{3}\]

Factored

\[\left(a - b\right) \cdot \left(a - c\right) \cdot \left(b - c\right) \cdot \left(a + b\right) \cdot \left(a + c\right) \cdot \left(b + c\right) + \left(a + b\right) \cdot \left(b + c\right) \cdot \left(a + c\right) \cdot \left(a a c + b b a + c c b\right) + a b \cdot c \cdot \left(a - b\right) \cdot \left(a - c\right) \cdot \left(b - c\right) = a^{4} b^{2} + a^{4} b c + a^{3} b^{3} + 3 a^{3} b^{2} c + a^{3} c^{2} b + a^{3} c^{3} + a^{2} b^{3} c + 3 a^{2} b^{2} c^{2} + 3 a^{2} c^{3} b + a^{2} c^{4} + b^{4} a c + 3 b^{3} c^{2} a + b^{2} c^{3} a + c^{4} a b + b^{4} c^{2} + b^{3} c^{3}\]

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