Question

( P+q+r=0 )
( begin{array}{l}p a q c r b p c & q a b b s a^{3} p q r-p^{3} a b c-q^{3} a b c+b^{3} p q s+c^{3} p q r-s^{3} a b cend{array} )
( =p q sleft(a^{3}+b^{3}+c^{3}right)-a b cleft(p^{3}+q^{3}+r^{3}right) theta )
As ( p+q+x=0 )
( Rightarrow p^{3}+q^{3}+r^{3}=-3 p q s )
(1) ( Rightarrow quad p q^{3}left(a^{3}+b^{3}+c^{3}right)+3 p q r a b c+b )
( =p q rleft[a^{3}+b^{3}+c^{3}+3 a b cright]-(2) )
Now, ( left|begin{array}{lll}a & b & c b & c & a c & a & bend{array}right|=-left(a^{3}+b^{3}+c^{3}+3 a b cright) )
So ( left|begin{array}{lll}a & b & c c & a & b b & c & aend{array}right|=a^{3}+b^{3}+c^{3}+3 a b c ) of row callse
change in sign
( becauseleft(2 Rightarrow p q &left|begin{array}{lll}a & b & c c & a & b b & c & aend{array}right|right. )

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Solution