αβ(α+β)+βγ(β+γ)+γα(γ+α)−6αβγ
You asked:
Evaluate the expression: \(α \cdot β \cdot \left(α + β\right) + β \cdot γ \cdot \left(β + γ\right) + γ \cdot α \cdot \left(γ + α\right) - 6 α β γ\)
MathBot Answer:
Evaluated
\(\displaystyle α \cdot β \cdot \left(α + β\right) + β \cdot γ \cdot \left(β + γ\right) + γ \cdot α \cdot \left(γ + α\right) - 6 α β γ = - 6 α β γ + α β \left(α + β\right) + α γ \left(α + γ\right) + β γ \left(β + γ\right) \)
Expanded
\[α \cdot β \cdot \left(α + β\right) + β \cdot γ \cdot \left(β + γ\right) + γ \cdot α \cdot \left(γ + α\right) - 6 α β γ = α^{2} β + α^{2} γ + β^{2} α - 6 α β γ + γ^{2} α + β^{2} γ + γ^{2} β\]
Factored
\[α \cdot β \cdot \left(α + β\right) + β \cdot γ \cdot \left(β + γ\right) + γ \cdot α \cdot \left(γ + α\right) - 6 α β γ = α^{2} β + α^{2} γ + β^{2} α - 6 α β γ + γ^{2} α + β^{2} γ + γ^{2} β\]