Question

varify the aerobric identity a3+b3=(a+b) (a2-ab + b2)​

Answer 1

Appropriate Question :-

Verify the algebraic identity ;

a³ + b³ = ( a + b ) ( a² - ab + b² )

Solution :-

Before starting the question , let's recall some algebraic identity which will help in the proof of the given identity ;

${ \quad \leadsto \quad { \pmb { \orange { \bf { (a+b)³ = a³+b³+3ab(a+b)}}}}}$

${ \quad \leadsto \quad { \pmb { \blue { \bf { (a+b)² = a² + b² + 2ab }}}}}$

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So , the identity that we have is ;

${\quad \leadsto \quad \sf (a+b)³ = a³ + b³ + 3ab(a+b)}$

can be further written as ;

${ : \implies \quad \sf a³ + b³ + 3ab(a+b) = ( a + b )³}$

Can be more further written as ;

${ : \implies \quad \sf a³ + b³ = ( a + b )³ - 3ab ( a + b ) }$

Can again written as ;

${ : \implies \quad \sf a³ + b³ = \underbrace{( a + b )}( a + b ) ( a + b ) - 3ab \underbrace{( a + b )} }$

Take ( a + b ) common;

${ : \implies \quad \sf a³ + b³ = ( a + b ) \bigg\{ ( a + b ) ( a + b ) - 3ab \bigg\}}$

Can be written as ;

${ : \implies \quad \sf a³ + b³ = ( a + b ) \bigg\{ ( a + b )² - 3ab \bigg\}}$

Using the above provided identity we have ;

${ : \implies \quad \sf a³ + b³ = ( a + b ) ( a² +b²+2ab - 3ab )}$

${ : \implies \quad \sf a³ + b³ = ( a + b ) ( a² +b²-ab )}$

${ : \implies \quad \bf a³ + b³ = ( a + b ) (a² -ab +b² )}$

$\quad { \bigstar { \underline { \boxed { \pmb { \bf { \red { \underbrace { \therefore a³ + b³ = ( a + b ) (a² -ab +b² ) }}}}}}}}{\bigstar}\quad \qquad$