Resolve into factors x²⁴-y²⁴
Answers
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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