If a,b,c are distinct, and $$111abca3b3c3$$ $$=(b$$−$$c)(c$$−$$a)(a$$−$$b)(a+b+c)$$ then Δ$$=111(x$$−$$a)2(x$$−$$b)2(x$$−$$c)2(x$$−$$b)(x$$−$$c)(x$$−$$c)(x$$−$$a)(x$$−$$a)(x$$−$$b)$$ vanishes if

Question

If a,b,c are distinct, and $$111abca3b3c3$$ $$=(b$$−$$c)(c$$−$$a)(a$$−$$b)(a+b+c)$$ then Δ$$=111(x$$−$$a)2(x$$−$$b)2(x$$−$$c)2(x$$−$$b)(x$$−$$c)(x$$−$$c)(x$$−$$a)(x$$−$$a)(x$$−$$b)$$ vanishes if

Infinity Learn · Accepted Answer

Multiplying $$C1$$ by (x-a), $$C2$$ by (x-b) and $$C3$$ by (x-c), we getΔ$$=1abcabca3b3c3abcabcabcWhere$$ $$A=x$$−a,$$B=x$$−b,$$C=x$$−cΔ$$=abca3b3c3111=($$−$$1)($$−$$1)111abca3b3c3$$                  $$=(b$$−$$c)(c$$−$$a)(a$$−$$b)(a+b+c)$$                  $$=(c$$−$$b)(a$$−$$b)(b$$−$$a)$$[$$3x$$−(a+b+c)]

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If a,b,c are distinct, and 111abca3b3c3 =(b−c)(c−a)(a−b)(a+b+c) then Δ=111(x−a)2(x−b)2(x−c)2(x−b)(x−c)(x−c)(x−a)(x−a)(x−b) vanishes if

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