Question

8x³ + 27y³ + 36x²y + 54x²y factories using identity ( class 9) here hard part is to find which identity we should use.. how to find identity in any expansion form ??​

Answer 1

GO THROUGH THE ATTACHED IMAGE..

Answer 2

Appropriate Question:-

➢ Factorize :

$\red{\rm :\longmapsto\: {8x}^{3} + {27y}^{3} + 36 {x}^{2}y + 54 {xy}^{2} }$

$\large\underline{\sf{Solution-}}$

Given expression is

$\rm :\longmapsto\: {8x}^{3} + {27y}^{3} + 36 {x}^{2}y + 54 {xy}^{2}$

can be rewritten as

$\rm \: = \: \: {(2x)}^{3} + {(3y)}^{3} + 3 \times {(2x)}^{2} \times 3y + 3 \times (2x) \times {(3y)}^{2}$

$\rm \: = \: \: {(2x)}^{3} + 3 \times {(2x)}^{2} \times 3y + 3 \times (2x) \times {(3y)}^{2} + {(3y)}^{3}$

We know that,

$\underbrace{\boxed{\bf{ {(x + y)}^{3} = {x}^{3} + 3 {x}^{2}y + {3xy}^{2} + {y}^{3}}}}$

$\rm \: = \: \: {(2x + 3y)}^{3}$

$\rm \: = \: \: (2x + 3y)(2x + 3y)(2x + 3y)$

Hence,

$\red{\rm :\longmapsto\: {8x}^{3} + {27y}^{3} + 36 {x}^{2}y + 54 {xy}^{2}}$

$\red{\rm \: = \: \: (2x + 3y)(2x + 3y)(2x + 3y)}$

Additional Information :-

➢ Let us take few more examples!!

➢ Factorize the following :-

$\red{\rm :\longmapsto\: {8x}^{3} - {27y}^{3} - 36 {x}^{2}y + 54 {xy}^{2} }$

can be rewritten as

$\rm \: = \: \: {(2x)}^{3} - {(3y)}^{3} - 3 \times {(2x)}^{2} \times 3y + 3 \times (2x) \times {(3y)}^{2}$

$\rm \: = \: \: {(2x)}^{3} - 3 \times {(2x)}^{2} \times 3y + 3 \times (2x) \times {(3y)}^{2} - {(3y)}^{3}$

We know that

$\underbrace{\boxed{\bf{ {(x - y)}^{3} = {x}^{3} - 3 {x}^{2}y + {3xy}^{2} - {y}^{3}}}}$

$\rm \: = \: \: {(2x - 3y)}^{3}$

$\rm \: = \: \: (2x - 3y)(2x - 3y)(2x - 3y)$

Hence,

$\red{\rm :\longmapsto\: {8x}^{3} - {27y}^{3} - 36 {x}^{2}y + 54 {xy}^{2} }$

$\red{\rm \: = \: \: (2x - 3y)(2x - 3y)(2x - 3y)}$

➢ Factorize the following :-

$\purple{\rm :\longmapsto\: {125x}^{3} + {8y}^{3} + 150 {x}^{2}y + 60{xy}^{2} }$

$\rm \: = \: \: {(5x)}^{3} + {(2y)}^{3} + 3 \times {(5x)}^{2} \times 2y + 3 \times (5x) \times {(2y)}^{2}$

$\rm \: = \: \: {(5x)}^{3} + 3 \times {(5x)}^{2} \times 2y + 3 \times (5x) \times {(2y)}^{2} + {(2y)}^{3}$

We know,

$\underbrace{\boxed{\bf{ {(x + y)}^{3} = {x}^{3} + 3 {x}^{2}y + {3xy}^{2} + {y}^{3}}}}$

$\rm \: = \: \: {(5x + 2y)}^{3}$

$\rm \: = \: \: (5x + 2y)(5x + 2y)(5x + 2y)$

Hence,

$\purple{\rm :\longmapsto\: {125x}^{3} + {8y}^{3} + 150 {x}^{2}y + 60{xy}^{2} }$

$\purple{\rm \: = \: \: (5x + 2y)(5x + 2y)(5x + 2y)}$