If | array ccc a+b+2 c & a & b c & b+c+2 a & b c & a & c+a+2 b array |=k( a+ b+ c)^3 , then (2 + - )^k is ( , , , k z⁺ )

Step-by-step solution

LHS = | array ccc a+b+2 c & a & b c & b+c+2 a & b c & a & c+a+2 b array | Operate: C ₁ C ₁+ C ₂+ C ₃= | array ccc 2(a+b+c) & a & b 2(a+b+c) & b+c+2 a & b 2(a+b+c) & a & c+a+2 b array |=2(a+b+c) | array ccc 1 & a & b 1 & b+c+2 a & b 1 & a & c+a+2 b array | Operate : R₂ R₂-R₁ ; R₃ R₃-R₁=2(a+b+c) | array ccc 1 & a & b 0 & (a+b+c) & 0 0 & 0 & (c+a+b) array | Expand by C ₁=2(a+b+c) .1 | array cc a+b+c & 0 0 & a+b+c array |=2(a+b+c)^3 k=2 and = = =1