Basics Faster Than Skills

Factorize (bc)6+(ca)6+(ab)69(ab)2(bc)2(ca)22(ab)3(ac)32(bc)3(ba)32(ca)3(cb)3(b-c)^{6}+(c-a)^{6}+(a-b)^{6}-9(a-b)^{2}(b-c)^{2}(c-a)^{2}-2(a-b)^{3}(a-c)^{3}-2(b-c)^{3}(b-a)^{3}-2(c-a)^{3}(c-b)^{3}

  • When you got the answer you must be surprised, feeling cheated.
  • Think about the title carefully, in that there are two ways to solve it.

Note by Jack Nicolson
3 years, 10 months ago

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0,same way as rohit.

Shreyash Rai - 3 years, 10 months ago

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Aha, another brilliant person

Jack Nicolson - 3 years, 10 months ago

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0

Ayush Pattnayak - 3 years, 10 months ago

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Yeah, did it take you a long time? :D

Jack Nicolson - 3 years, 10 months ago

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My advice to all regarding such long questions is to first try your luck with 0,and if it doesn't work,move to the next one

Ayush Pattnayak - 3 years, 10 months ago

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@Ayush Pattnayak Ha-ha, next time I'll try it, thanks for your nice method :D

Jack Nicolson - 3 years, 10 months ago

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Certainly not....

Ayush Pattnayak - 3 years, 10 months ago

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Yes indeed...it looks quite symmetric doesn't it?

Let ab=x,bc=y,ca=za-b=x, b-c=y, c-a=z

Then the expression becomes:

x6+y6+z69x2y2z2+2x3y3+2y3z3+2z3x3x^6+y^6+z^6-9x^2y^2z^2+2x^3y^3+2y^3z^3+2z^3x^3

=(x3+y3+z3)2(3xyz)2= (x^3+y^3+z^3)^2-(3xyz)^2

=(x3+y3+z33xyz)(x3+y3+z3+3xyz)= (x^3+y^3+z^3-3xyz)(x^3+y^3+z^3+3xyz)

Now we note that x+y+z=0x+y+z=0 which means that x3+y3+z3=3xyzx^3+y^3+z^3=3xyz and our whole expression becomes 0

Rohit Sachdeva - 3 years, 10 months ago

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That's why I said you'll be shocked :D, your method is faster than those I thought because I, in fact, want you guys to get it simplified step by step first, obviously you didn't jump into my trap :P

Jack Nicolson - 3 years, 10 months ago

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