Math

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if alpha and beta are the zeroes of the polynomial f(x)=x^2-3x-2 , find a quadratic polynomial whose zeroes are 1/2(alpha)+beta and 1/2(beta)+alpha ?

Please... i have no idea !!

  • Math -

    The zeroes of the polynomial x ^ 2 - 3 x - 2 are


    [ 3 - sqrt ( 17 ) ] / 2

    [ 3 + sqrt ( 17 ) ] / 2

    so:

    alpha = [ 3 - sqrt ( 17 ) ] / 2

    beta = [ 3 + sqrt ( 17 ) ] / 2



    x1 = alpha / 2 + beta =

    [ 3 - sqrt ( 17 ) ] / ( 2 * 2 ) + [ 3 + sqrt ( 17 ) ] / 2 =

    [ 3 - sqrt ( 17 ) ] / 4 + 2 * [ 3 + sqrt ( 17 ) ] / ( 2 * 2 ) ] =

    [ 3 - sqrt ( 17 ) ] / 4 + 2 * [ 3 + sqrt ( 17 ) ] / ( 2 * 2 ) =

    [ 3 - sqrt ( 17 ) ] / 4 + [ 6 + 2 sqrt ( 17 ) ] / 4 =

    [ 3 - sqrt ( 17 ) + 6 + 2 sqrt ( 17 ) ] / 4 =

    [ sqrt ( 17 ) + 9 ] / 4



    beta / 2 + alpha = alpha / 2 + beta =

    [ 3 + sqrt ( 17 ) ] / ( 2 * 2 ) + [ 3 - sqrt ( 17 ) ] / 2 =

    [ 3 + sqrt ( 17 ) ] / 4 + 2 * [ 3 - sqrt ( 17 ) ] / ( 2 * 2 ) ] =

    [ 3 + sqrt ( 17 ) ] / 4 + 2 * [ 3 - sqrt ( 17 ) ] / ( 2 * 2 ) =

    [ 3 + sqrt ( 17 ) ] / 4 + [ 6 - 2 sqrt ( 17 ) ] / 4 =

    [ 3 + sqrt ( 17 ) + 6 - 2 sqrt ( 17 ) ] / 4 =

    [ - sqrt ( 17 ) + 9 ] / 4


    Now you must use Lagrange resolvents:

    y = a x ^ 2 + b x + c = a ( x - x1 ) ( x - x2 )

    in this case a = 1 so :

    y = ( x - x1 ) ( x - x2 )

    y = ( 1 / 4 )[ sqrt ( 17 ) + 9 ] * ( 1 / 4 )[ - sqrt ( 17 ) + 9 ]

    y = [ x ^ 2 - 18 x + 64 ] / 16

    y = x ^ 2 / 16 - 9 x / 8 + 4

  • Math -

    x2 = beta / 2 + alpha =

    [ 3 + sqrt ( 17 ) ] / ( 2 * 2 ) + [ 3 - sqrt ( 17 ) ] / 2 =

    [ 3 + sqrt ( 17 ) ] / 4 + 2 * [ 3 - sqrt ( 17 ) ] / ( 2 * 2 ) ] =

    [ 3 + sqrt ( 17 ) ] / 4 + 2 * [ 3 - sqrt ( 17 ) ] / ( 2 * 2 ) =

    [ 3 + sqrt ( 17 ) ] / 4 + [ 6 - 2 sqrt ( 17 ) ] / 4 =

    [ 3 + sqrt ( 17 ) + 6 - 2 sqrt ( 17 ) ] / 4 =

    [ - sqrt ( 17 ) + 9 ] / 4

  • Math -

    Thanks :) I understood (Y)

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