Resolve x^18- y^18 into five factors.

1.	Resolve x^18- y^18 into five factors.
Answer» ANSWER: ( x + y ) ( x - y ) ( x² + y² - xy ) ( x² + y² + xy ) ( x⁶ + y⁶ - x³y³ ) ( x⁶ + y⁶ + x³y³ ) Step-by-step explanation: Given----> x¹⁸ - y¹⁸ To find ----> FACTORS of given expression Solution-----> We know that, 1) a² - b² = ( a + b ) ( a - b ) 2) a³ + b³ = ( a + b ) ( a² + b² - ab ) 3) a³ - b³ = ( a - b ) ( a² + b² + ab ) Now, x¹⁸ - y¹⁸ = ( x⁹ )² - ( y⁹ )² Applying first IDENTITY , we get , = ( x⁹ + y⁹ ) ( x⁹ - y⁹ ) = { ( x³)³+( y³ )³ } { ( x³ )³ - ( y³ )³ } Applying second and THIRD identity , we get , = ( x³ + y³ ) { ( x³)² + ( y³ )² - x³y³ } ( x³ - y³ ) { ( x³ )² + ( y³ )² + x³ y³ } Applying second and third identity again , we get = ( x + y ) ( x² + y² - xy ) ( x⁶ + y⁶ - x³ y³ ) ( x - y ) ( x² + y² + xy ) ( x⁶ + y⁶ + x³y³ ) = ( x + y ) ( x - y ) ( x² + y² - xy ) ( x² + y² - xy ) ( x⁶ + y⁶ - x³y³ ) ( x⁶ + y⁶ + x⁶ y⁶ )

Answer»

( x + y ) ( x - y ) ( x² + y² - xy ) ( x² + y² + xy ) ( x⁶ + y⁶ - x³y³ ) ( x⁶ + y⁶ + x³y³ )

Step-by-step explanation:

Given----> x¹⁸ - y¹⁸

To find ----> FACTORS of given expression

Solution-----> We know that,

1) a² - b² = ( a + b ) ( a - b )

2) a³ + b³ = ( a + b ) ( a² + b² - ab )

3) a³ - b³ = ( a - b ) ( a² + b² + ab )

Now,

x¹⁸ - y¹⁸ = ( x⁹ )² - ( y⁹ )²

Applying first IDENTITY , we get ,

= ( x⁹ + y⁹ ) ( x⁹ - y⁹ )

= { ( x³)³+( y³ )³ } { ( x³ )³ - ( y³ )³ }

Applying second and THIRD identity , we get ,

= ( x³ + y³ ) { ( x³)² + ( y³ )² - x³y³ } ( x³ - y³ ) { ( x³ )² +

( y³ )² + x³ y³ }

Applying second and third identity again , we get

= ( x + y ) ( x² + y² - xy ) ( x⁶ + y⁶ - x³ y³ ) ( x - y ) ( x² + y² + xy ) ( x⁶ + y⁶ + x³y³ )

= ( x + y ) ( x - y ) ( x² + y² - xy ) ( x² + y² - xy ) ( x⁶ + y⁶ - x³y³ ) ( x⁶ + y⁶ + x⁶ y⁶ )

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