Question

A+b=1 a cube plus b cube then a square plus b square is what

Answer 1

(a+b)³ = (a+b)²(a+b)

(a+b)³ = (a² + b² + 2ab)(a+b)

(a+b)³ = a(a² + b² + 2ab) +b(a² + b² + 2ab)

(a+b)³ = a³ + ab² + 2a²b + a²b + b³ +2ab²

(a+b)³ = a³ + b³ + 3ab² + 3a²b

We have find a³ + b³. therefore, transfer other terms to one side:

(a+b)³ - 3a²b - 3a²b = a³ + b³

a³ + b³ = (a+b)³ - 3a²b - 3a²b

a³ + b³ = (a+b)³ -3ab(a+b)

Taking (a+b) common,

a³ + b³ = (a+b){(a+b)² -3ab}

a³ + b³ = (a+b){a² + b² + 2ab -3ab}

a³ + b³ = (a+b)(a² + b² - ab)

Hence proved;

Note:-

(a+b)² = (a+b)(a+b)

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

= a² + ab + ba + b²

= a² + b² + 2ab

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