Question

x⁸ + x⁴y⁴ + y⁸ ÷ x⁴ + x²y² + y⁴
please solve this fast. quickly​

Answer 1

Solution :

Here , we need to divide the numerator, x⁸ + x⁴y⁴ + y⁸ by the denominator , x⁴ + x²y² + y⁴

Let us simplify the numerator first .

x⁸ + x⁴y⁴ + y⁸

=> x⁸ + x⁴y⁴ + x⁴ y⁴ + y⁸ - x⁴ y⁴

=> [ x⁴ ] ² + 2 [ x⁴ ][ y⁴ ] + [ y⁴ ]² - x⁴ y⁴

=> [ x⁴ + y⁴ ]² - x⁴ y⁴

=> [ x⁴ + y⁴ ]² - [ x² y² ]²

=> [ x⁴ + y⁴ + x² y² ][ x⁴ + y⁴ - x² y² ]

( x⁸ + x⁴y⁴ + y⁸ ) / ( x⁴ + x²y² + y⁴ )

=> [ x⁴ + y⁴ + x² y² ][ x⁴ + y⁴ - x² y² ] / ( x⁴ + x²y² + y⁴ )

=> [ x⁴ + y⁴ - x² y² ]

=> [ x⁴ - x² y² + y⁴ ]

Answer :

Quotient = [ x⁴ - x² y² + y⁴ ]

Remainder = 0 .

Additional Information :

(a + b)² = a² + 2ab + b²

(a + b)² = (a - b)² + 4ab

(a - b)² = a² - 2ab + b²

(a - b)² = (a + b)² - 4ab

a² + b² = (a + b)² - 2ab

a² + b² = (a - b)² + 2ab

2 (a² + b²) = (a + b)² + (a - b)²

4ab = (a + b)² - (a - b)²

ab = {(a + b)/2}² - {(a-b)/2}²

(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

(a + b)³ = a³ + 3a²b + 3ab² b³

(a + b)³ = a³ + b³ + 3ab(a + b)

(a - b)³ = a³ - 3a²b + 3ab² - b³

a³ + b³ = (a + b)( a² - ab + b² )

a³ + b³ = (a + b)³ - 3ab( a + b)

a³ - b³ = (a - b)( a² + ab + b²)

a³ - b³ = (a - b)³ + 3ab ( a - b )

_____________________________

Answer 2

Solution:

x⁸ + x⁴y⁴ + y⁸ ÷ x⁴ + x²y² + y⁴

Solution:
x8+x4y4+y8 / x4+x2y2+y4

= x8+x4*y4+y8 / x4+x2*y2+y4

= (x2+y2+xy)(x2+y2-xy)(x4+y4-x2y2) /
(x2+y2+xy)(x2+y2-xy)

Now cancel the common factor (x2+y2+xy)(x2+y2-xy)

= (x4+y4-x2y2)

Or simple subtract the exponent of same terms(shortcut form)

x8-x4=x4
x4y4-x2y2=x2y2
y8-y4=y4

= (x4-x2y2+y4)

Hope this will help.