Question

Verify identity

a3
+ b3
= (a + b) (a2
- ab + b2
)
Please answer in LHS = RHS format
​

Answer 1

Answer:

You know that,

(a + b)³ = a³ + 3ab(a + b) + b³

then,

a³ + b³ = (a + b)³ – 3ab(a + b)

= (a + b)[(a + b)² – 3ab]

= (a + b)(a² + 2ab + b² – 3ab)

= (a + b)(a² – ab + b² )

Answer 2

Since the expression is derived from (a+b)^3

So let us expand it

(a+b)^3

= (a+b) (a+b) (a+b)

={(a+b) (a+b)} (a+b)

={a(a+b) + b(a+b)} (a+b)

=(a^2 + ab + ab + b^2) (a+b)

=(a^2 + b^2 + 2ab) (a+b)

=a^2(a+b) + b^2(a+b) + 2ab(a+b)

=a^3 + a^2b + ab^2 + b^3 + 2a^2b + 2ab^2

=a^3 + b^3 + 3a^2b + 3ab^2

=a^3 + b^3 + 3ab(a+b)

Now when we have expanded (a+b)^3 = a^3 + b^3 + 3ab(a+b)

We can equate it

(a+b)^3 = a^3 + b^3 + 3ab(a+b)

(a+b)^3 - 3ab(a+b) = a^3 + b^3

a^3 + b^3 = (a+b)^3 - 3ab(a+b)