Question

Resolve into factors: x²⁴-y²⁴​

Answer 1

Given : x²⁴-y²⁴

To Find : Factorize

Solution:

x²⁴-y²⁴

(xᵃ)ᵇ = xᵃᵇ

24 = 12 * 2

= (x¹²)² - (y¹²)²

using a² - b² = (a + b)(a - b)

= (x¹² + y¹²)  (x¹² - y¹²)

12 = 6 * 2

= (x¹² + y¹²)  ((x⁶)² - (y⁶)²)

again using a² - b² = (a + b)(a - b)

=  (x¹² + y¹²) ( x⁶ +  y⁶ )  ( x⁶  -  y⁶ )

6 = 3 x 2

=  (x¹² + y¹²) ( x⁶ +  y⁶ )  ( (x³)²  -  (y³)²)

again using a² - b² = (a + b)(a - b)

=  (x¹² + y¹²) ( x⁶ +  y⁶ )(x³  +  y³ )(x³ -   y³ )

Using a³ - b³ = (a - b)(a² + b² + ab)

= (x¹² + y¹²) ( x⁶ +  y⁶ )(x³  +  y³ )(x -  y)(x² + y² + xy)

 = (x¹² + y¹²) ( x⁶ +  y⁶ )(x³  +  y³ )(x² + y² + xy)(x -  y)

Using a³ + b³ = (a + b)(a² + b² - ab)

 = (x¹² + y¹²) ( x⁶ +  y⁶ )(x² + y² - xy)(x +  y)(x² + y² + xy)(x -  y)

x²⁴-y²⁴  = (x¹² + y¹²) ( x⁶ +  y⁶ )(x² + y² - xy)(x +  y)(x² + y² + xy)(x -  y)

x¹² + y¹²   can be further factorizes as (x⁴)³ + (y⁴)³  Using a³ + b³ = (a + b)(a² + b² - ab)

= (x⁴ + y⁴)(x⁸ + y⁸ - x⁴y⁴)

x⁶ + y⁶   can be further factorizes as (x²)³ + (y²)³ Using a³ + b³ = (a + b)(a² + b² - ab)

= (x² + y²)(x⁴ + y⁴ - x²y²)

x²⁴-y²⁴  =  (x⁴ + y⁴)(x⁸ + y⁸ - x⁴y⁴)(x² + y²)(x⁴ + y⁴ - x²y²)(x² + y² - xy)(x +  y)(x² + y² + xy)(x -  y)

x²⁴-y²⁴  =  (x +  y)(x -  y)(x² + y²)(x² + y² - xy)(x² + y² + xy) (x⁴ + y⁴)(x⁴ + y⁴ - x²y²)(x⁸ + y⁸ - x⁴y⁴)

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