Prove: `|(1,b+c ,b^2+c^2),( 1,c+a ,c^2+a^2),( 1,a+b ,a^2+b^2)|=(a-b)(b-c)(c-a)`

Explanation

`{ Let LHS }=Delta=|(1,b+c ,b^2+c^2),( 1,c+a ,c^2+a^2),( 1,a+b ,a^2+b^2)|`


`Delta=[[0,(b+c)-(c+a),(b^2+c^2)-(c^2+a^2)],[0,(c+a)-(a+b),(c^2+a^2)-(a^2+b^2)],[0,(a+b),(a^2+b^2)]]``{`Applying `underset(1)(R)->underset(1)(R)-underset(2)(R)` And `underset(2)(R)->underset(2)(R)-underset(3)(R)}`



`implies [[0,b-a,(b^2-a^2)],[0,(c-b),(c^2-b^2)],[0,(a+b),(a^2+b^2)]]`



`{`Taking `-1` common from `underset(1)(R)` and `underset(2)(R)}`


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