Question

(a+b+c)^2 =
ans me and
a^3-b^3=​

Answer 1

Answer:

• (a + b + c )² = a² + b² + c² + 2ab + 2bc + 2ca
• a³ -b³ = (a – b)(a² + b² + ab)

Explanation:

☆ The proof of the identities is as follows:

1️⃣To Prove: (a + b + c )² = a² + b² + c² + 2ab + 2bc + 2ca

Proof:

⋆ LHS

= (a + b + c )²

= (a + b + c ) × (a + b + c )

= a (a + b + c ) + b (a + b + c ) + c (a + b + c )                [Distributive property]

= (a×a) + (a×b) + (a×c) + (b×a)+ (b×b) + (b×c) + (c×a) + (c×b) + (c×c)

= a² + ab + ac + ba + b² + bc + ca + cb + c²

= a² + b² + c² + (ab+ ab) + (bc + bc) + (ca + ca)

= a² + b² + c² + 2ab + 2bc + 2ca = RHS

Hence, LHS = RHS, proved.

2️⃣To prove: a³ -b³ = (a – b)(a² + b² + ab)

Proof:

⋆ RHS

= (a – b)(a² + b² + ab)

= a (a² + b² + ab) –  [b (a² + b² + ab)]                             [Distributive property]

= (a × a²) + ( a× b² ) + (a × ab) – [ (b × a²) + (b × b²) + ( b × ab)  

= a³ + ab² + a²b –  ( a²b + b³ + ab² )

= a³ +  ab² + a²b –a²b –b³ – ab²

= a³ – b³ + ab² –ab² + a²b – a²b

= a³ – b³ = LHS

Hence, LHS = RHS, proved.

Hope you understood.

Thank You.