Question

Factories x^12 - y^12​

Answer 1

Answer:

( x - y ) ( x + y ) ( x² - xy + y² ) ( x² + xy + y² ) ( x² + y² ) ( x⁴ - x²y² + y⁴ )

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Step-by-step explanation:

x¹² - y¹²

= (x⁶)² - (y⁶)²

= ( x⁶ - y⁶ ) ( x⁶ + y⁶ )             (1)

To avoid getting muddled, deal with these factors one at a time.

x⁶ - y⁶

= (x³)² - (y³)²

= ( x³ - y³ ) ( x³ + y³ )           (2)

Now we need to recall:

x³ - y³ = ( x - y ) ( x² + xy + y² )

x³ + y³ = ( x + y ) ( x² - xy + y² )

Putting these together into (2), we have

x⁶ - y⁶ = ( x - y ) ( x + y ) ( x² - xy + y² ) ( x² + xy + y² )      (3)

Now the second factor in (1) is

x⁶ + y⁶

= (x²)³ + (y²)³

= ( x² + y² ) ( (x²)² - x²y² + (y²)² )

= ( x² + y² ) ( x⁴ - x²y² + y⁴ )               (4)

Putting (3) and (4) together into (1), we finally have

x¹² - y¹²

= ( x - y ) ( x + y ) ( x² - xy + y² ) ( x² + xy + y² ) ( x² + y² ) ( x⁴ - x²y² + y⁴ )

Answer 2

a² - b² =( a + b ) ( a - b )

a³ - b³ = ( a - b ) ( a² + a + b² )

$\sf x^{12}-y^{12}$

$\sf (x^{6}-y^{6})(x^{6}+y^{6})$

$\sf (x^{3}-y^{3})(x^{3}+y^{3}) (x^{6}+y^{6})$

$\sf (x-y)(x^{2} +xy+y^{2} )(x^{3}+y^{3}) (x^{6}+y^{6})$