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Prove that : (a+b)^3+(b+c)^3+(c+a)^3-3(a...

Prove that : `(a+b)^3+(b+c)^3+(c+a)^3-3(a+b)(b+c)(c+a)=2(a^3+b^3+c^3-3a b c)`

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LHS:
`=a^3+b^3+3a^2b+3ab^2+b^3+c^3+3b^2c+3bc^2+c^3+a^3+3c^2a+3ca^2-3abc-3ac^2-3b^2c-3bc^2-3a^2b-3a^2c-3ab^2-3abc`
`=2a^3+2b^3+2c^3-6abc`
`=2(a^3+b^3+c^3-3abc)`
Hence proved.
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RD SHARMA-FACTORIZATION OF ALGEBRAIC EXPRESSIONS-All Questions
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  2. Factorize each of the following expressions: 2p(a-b)+3q(5a-5b)+4r(2a...

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  3. Prove that : (a+b)^3+(b+c)^3+(c+a)^3-3(a+b)(b+c)(c+a)=2(a^3+b^3+c^3-3a...

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  4. Factorize the following expression: 4a^2+12 a b+9b^2-8a-12 b

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  5. Factorisation by grouping the terms: (x^2+3x)^2-5(x^2+3x)-y(x^2+3x)+5y

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  6. Factorize each of the following expressions : 4a^2-9b^2-2a-3b

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  7. Factorize each of the following algebraic expressions : x^8 - y^8 a...

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  8. Factorize: (i) 8x^3+27 y^3+z^3-18 x y z, (ii) a^3-8b^3-64 c^3-24 a...

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  9. Factorize : 64 a^3+125 b^3+240 a^2b+300 a b^2 8/(27)x^3+1+4/3x^2+2x...

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  10. Examples based on the middle term concepts : (i) 4sqrt3 x^2 + 5x - 2sq...

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  11. Factorize : (x^2-4x)(x^2-4x-1)-20

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  12. Factorize : p^3(q-r)^3+q^3(r-p)^3+r^3(p-q)^3

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  13. Factorize : (x-y)^3+(y-z)^3+(z-x)^3

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  14. Factorize : 2sqrt(2)x^3+3sqrt(3)y^3+sqrt(5)(5-3sqrt(6)x y)

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  15. Prove that : a^3+b^3+c^3-3a b c=1/2(a+b+c){(a-b)^2+(b-c)^2+(c-a)^2}

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  16. Factorize : 2sqrt(2)a^3+8b^3-27 c^3+18sqrt(2)a b c

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  17. Resolve a^3-b^3+1+3a b into factors.

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  18. Factorise : (a+b)^3+(b+c)^3+(c+a)^3-3(a+b)(b+c)(c+a)

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  19. Factorize each of the following expressions : x^2+2x y+y^2-a^2+2a b...

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  20. Factorize each of the following expressions : 3-12(a-b)^2

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