Home
Class 9
MATHS
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)`

Text Solution

Verified by Experts

let LHS
`=(a+b)^3+(b+c)^3+(c+a)^3-3(a+b)(b+c)(c+a)`
we know, `[x^3+y^3+z^3-3xyz=(x+y+z)(x^2+y^2+z^2-xy-yz-zx)]`
so,
`=((a+b)+(b+c)+(c+a))((a+b)^2+(b+c)^2+(c+a)^2-(a+b)(b+c)+(b+c)(c+a)+(c+a)(a+b))`
`=2(a+b+c)(a^2+b^2+c^2-ab-bc-ca)`
`=2(a^3+b^3+c^3-3abc)=`RHS
Promotional Banner

Topper's Solved these Questions

  • EXPONENTS OF REAL NUMBER

    RD SHARMA|Exercise All Questions|186 Videos

  • FACTORIZATION OF POLYNOMIAL

    RD SHARMA|Exercise All Questions|220 Videos

Similar Questions

Explore conceptually related problems

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)-3abc)

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

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

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

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

Prove that (a+b+c)^(3) =a^(3)+b^(3)+c^(3)+3(b+c) (c+a) (a+b)

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

Prove that |(a+b,b+c,c+a),(b+c,c+a,a+b),(c+a,a+b,b+c)|=- 2 (a^3+ b^3+c^3-3abc)

If 2s=a+b+c, prove that (s-a)^(3)+(s-b)^(3)+(s-c)^(3)-3(s-a)(s-b)(s-c)=(1)/(2)(a^(3)+b^(3)+c^(3)-3abc)

a^3(b-c)^3+b^3(c-a)^3+c^3(a-b)^3