f.3 maths

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問： Factorization : x^3 - 7x + 6

解：

x^3 - 7x + 6

= x^3 - x^2 + x^2 - x - 6x + 6

= x^2 ( x - 1 ) + x ( x - 1 ) - 6 ( x - 1 )

= ( x - 1 ) ( x^2 + x - 6 )

= ( x - 1 ) ( x^2 - 2x + 3x - 6 )

= ( x - 1 ) [ x ( x - 2 ) + 3 ( x - 2 ) ]

= ( x - 1 ) ( x - 2 ) ( x + 3 )

答：x^3 - 7x + 6 = ( x - 1 ) ( x - 2 ) ( x + 3 )

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問： 化簡 { x^(-1) - y^(-1) } / { x^(-2) - y ^(-2) }

解：

[ x^(-1) - y^(-1) ] / [ x^(-2) - y ^(-2) ]

= ( 1/x - 1/y ) / [ (1/x)^2 - (1/y)^2 ]

= ( 1/x - 1/y ) / [ ( 1/x + 1/y )( 1/x - 1/y ) ]

= 1 / ( 1/x + 1/y )

= xy / ( x + y )

答：[ x^(-1) - y^(-1) ] / [ x^(-2) - y ^(-2) ] = xy / ( x + y )

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