Simplest algebra

Algebra Level 2

( a − b ) 3 + ( b − c ) 3 + ( c − a ) 3 ( a − b ) ( b − c ) ( c − a ) \frac{ (a-b)^3+(b-c)^3+(c-a)^3} {(a-b)(b-c)(c-a) }

What is the value of the expression above if a ≠ b , b ≠ c , a ≠ c a \ne b, b \ne c, a \ne c .

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Chaitnya Shrivastava by Chaitnya Shrivastava , 20, India

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1 solution

Parth Lohomi
Apr 9, 2015

Upvote if you are satisfied!!

Using : If m + n + l = 0 m+n+l=0 then m 3 + n 3 + l 3 = 3 m n l m^3+n^3+l^3=3mnl

We see that ( a − b ) + ( b − c ) + ( c − a ) = 0 (a-b)+(b-c)+(c-a)=0 so ( a − b ) 3 + ( c − a ) 3 + ( b − c ) 3 = 3 ( a − b ) ( b − c ) ( c − a ) (a-b)^3+(c-a)^3+(b-c)^3 =3(a-b)(b-c)(c-a)

So the equation converts into

3 ( a − b ) ( b − c ) ( c − a ) ( a − b ) ( b − c ) ( c − a ) = 3 \dfrac{3(a-b)(b-c)(c-a)}{(a-b)(b-c)(c-a)} = \boxed{3}

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Parth Lohomi Isn't this question the same as this one

Abdur Rehman Zahid - 6 years, 2 months ago

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Of Course it is... but you see many easy questions are repeated ..so it is OK ⌣ ¨ \ddot\smile

Parth Lohomi - 6 years, 2 months ago

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