SOLVE i) (-2015)^3+(-2016)^3+(4031)^3
ii) a(b-c)^2+b(c-a)^2+c(a-b)^2+8abc
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1)
![-2015^3-2016^3+4031^3\\\\=-(2015^3+2016^3)+4031^3\\\\=-(2015+2016)(2015^2-2015*2016+2016^2)+4031[2015+2016]^2\\\\-4031[ 2015^2-2015*2016+2016^2]\\.\ \ \ \ \ +4031[ 2015^2+2*2015*2016+2016^2]\\\\=4031*3*2015*2016\\\\=49,124,668,320\\](https://tex.z-dn.net/?f=-2015%5E3-2016%5E3%2B4031%5E3%5C%5C%5C%5C%3D-%282015%5E3%2B2016%5E3%29%2B4031%5E3%5C%5C%5C%5C%3D-%282015%2B2016%29%282015%5E2-2015%2A2016%2B2016%5E2%29%2B4031%5B2015%2B2016%5D%5E2%5C%5C%5C%5C-4031%5B+2015%5E2-2015%2A2016%2B2016%5E2%5D%5C%5C.%5C+%5C+%5C+%5C+%5C+%2B4031%5B+2015%5E2%2B2%2A2015%2A2016%2B2016%5E2%5D%5C%5C%5C%5C%3D4031%2A3%2A2015%2A2016%5C%5C%5C%5C%3D49%2C124%2C668%2C320%5C%5C "-2015^3-2016^3+4031^3\\\\=-(2015^3+2016^3)+4031^3\\\\=-(2015+2016)(2015^2-2015*2016+2016^2)+4031[2015+2016]^2\\\\-4031[ 2015^2-2015*2016+2016^2]\\.\ \ \ \ \ +4031[ 2015^2+2*2015*2016+2016^2]\\\\=4031*3*2015*2016\\\\=49,124,668,320\\")
2)
![a(b-c)^2+b(c-a)^2+c(a-b)^2+8abc\\\\=ab^2+ac^2-2abc+bc^2+ba^2-2abc+c(a-b)^2+8abc\\\\=ab(a+b)+c^2(a+b)+4abc+c(a-b)^2\\\\=(a+b)(ab+c^2)+4abc+ca^2+cb^2-2abc\\\\=(a+b)(ab+c^2)+c(a+b)^2\\\\(a+b)[ab+c^2+ca+cb]\\\\=(a+b)(b+c)(c+a)\\](https://tex.z-dn.net/?f=a%28b-c%29%5E2%2Bb%28c-a%29%5E2%2Bc%28a-b%29%5E2%2B8abc%5C%5C%5C%5C%3Dab%5E2%2Bac%5E2-2abc%2Bbc%5E2%2Bba%5E2-2abc%2Bc%28a-b%29%5E2%2B8abc%5C%5C%5C%5C%3Dab%28a%2Bb%29%2Bc%5E2%28a%2Bb%29%2B4abc%2Bc%28a-b%29%5E2%5C%5C%5C%5C%3D%28a%2Bb%29%28ab%2Bc%5E2%29%2B4abc%2Bca%5E2%2Bcb%5E2-2abc%5C%5C%5C%5C%3D%28a%2Bb%29%28ab%2Bc%5E2%29%2Bc%28a%2Bb%29%5E2%5C%5C%5C%5C%28a%2Bb%29%5Bab%2Bc%5E2%2Bca%2Bcb%5D%5C%5C%5C%5C%3D%28a%2Bb%29%28b%2Bc%29%28c%2Ba%29%5C%5C "a(b-c)^2+b(c-a)^2+c(a-b)^2+8abc\\\\=ab^2+ac^2-2abc+bc^2+ba^2-2abc+c(a-b)^2+8abc\\\\=ab(a+b)+c^2(a+b)+4abc+c(a-b)^2\\\\=(a+b)(ab+c^2)+4abc+ca^2+cb^2-2abc\\\\=(a+b)(ab+c^2)+c(a+b)^2\\\\(a+b)[ab+c^2+ca+cb]\\\\=(a+b)(b+c)(c+a)\\")
2)
kvnmurty:
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