Question

the gcd of x4-y4 and x2-y2 is ​

Answer 1

SOLUTION

TO DETERMINE

The GCD of

$\sf{ {x}^{4} - {y}^{4} \: \: \: \: and \: \: \: {x}^{2} - {y}^{2} }$

EVALUATION

We have to find the GCD of

$\sf{ {x}^{4} - {y}^{4} \: \: \: \: and \: \: \: {x}^{2} - {y}^{2} }$

Now

$\sf{ ({x}^{4} - {y}^{4}) =( {x}^{2} + {y}^{2} )( {x}^{2} - {y}^{2}) = ( {x}^{2} + {y}^{2} )(x + y)(x - y)}$

$\sf{ {x}^{2} - {y}^{2} = (x + y)(x - y) }$

Hence the required GCD

$\sf{ = (x + y)(x - y)}$

$= \sf{ {x}^{2} - {y}^{2} }$

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