Question

find the HCF of the following polynomial : x8 - y8 ; ( x4 - y4) (X + y)​

Answer 1

SOLUTION

TO DETERMINE

The HCF of the polynomials

$\sf{ {x}^{8} - {y}^{8} \: \: \: and \: \: ( {x}^{4} - {y}^{4} )(x + y)}$

FORMULA TO BE IMPLEMENTED

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

EVALUATION

Here the given two polynomials are

$\sf{ {x}^{8} - {y}^{8} \: \: \: and \: \: ( {x}^{4} - {y}^{4} )(x + y)}$

Now

$\sf{ {x}^{8} - {y}^{8} }$

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

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

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

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

Again

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

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

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

Hence required HCF

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

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

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

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