126. If \(\Delta\) is defined for integers \(a,b\) by \(a\,\Delta\,b = a\times b − 2ab + b^2\):
(i) Compute \(4\,\Delta\,(−3)\) and \((−3)\,\Delta\,4\). Are they equal?
(ii) Compute \((−7)\,\Delta\,(−1)\) and \((−1)\,\Delta\,(−7)\). Are they equal?
(i) 4 Δ (−3) = 21, (−3) Δ 4 = 28, No
(ii) (−7) Δ (−1) = −6, (−1) Δ (−7) = 42, No
Plug into \(ab−2ab+b^2\):
(i) \(4,−3\): \(−12−(−24)+9=21\); swapping gives \(−12−(−24)+16=28\).
(ii) \(−7,−1\): \(7−14+1=−6\); swapping gives \(7−14+49=42\). Hence not commutative.