Question

(a + b + C )ka hole 3
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Answer 1

Question:

(a+b+c)³ = ?

Solution:

(a + b + c)³ = a³ + b³ + c³ + 3 (a +b) (b + c) (a+ c)

Explanation :

We know that

(a+b+c)² = a² +b² + c²+2ab+2bc+2ca

=a² +b² + c²+2(ab+bc+ca)

Now,

(a+b+c)³ can be written as (a+b+c)² (a+b+c)

so,

=> (a+b+c)³= (a+b+c)² (a+b+c)

=> (a+b+c)³= [a² +b² + c²+2(ab+bc+ca)] (a+b+c)

In the above step we have expanded (a+b+c)² using identity as above mentioned.

= a²[a+b+c] +b²[a+b+c]+c²[a+b+c] +2[ (ab+bc+ca)[a+b+c] ]

= a³+a²b+a²c + b²a+b³+b²c + c²a +c²b +c³ + 2ab[a+b+c] +2bc(a+b+c) +2ca(a+b+c)

=a³+a²b+a²c + b²a+b³+b²c + c²a +c²b +c³ +2a²b+2ab²+2abc +2abc+2b²c +2bc² +2a²c +2abc +2a²c

=a³+b³+c³+6abc +a²b+2a²b+a²c + b²a+b²c+2ab² +c²a+c²b+2c²b

=a³+b³+c³+6abc+3a²[b+c] +3b²(a+c) +3c²(a+b)

=a³+b³+c³+6abc+3[a²[b+c] +b²(a+c) +c²(a+b)]

on further simplifying, we get

(a + b + c)³= a³ + b³ + c³ + 3 [(a +b) (b + c) (a+ c)]

Hope it helps you.