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Thread: help me

  1. #1
    Junior Member
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    help me

    A Polynomial of lowest degree with integral coefficient, whose one of the Zero is root3+root2 is
    a) a(x^4-6x^2+10
    b) a(x^4-10x^2+1)
    c) a(x^4-9x^2+1)
    d) none of these
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  2. #2
    MHF Contributor red_dog's Avatar
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    If $\displaystyle x_1=\sqrt{3}+\sqrt{2}$ then $\displaystyle x_2=\sqrt{3}-\sqrt{2}, \ x_3=-\sqrt{3}+\sqrt{2}, \ x_4=-\sqrt{3}-\sqrt{2}$

    Then $\displaystyle P(x)=a(x-x_1)(x-x_2)(x-x_3)(x-x_4)=$

    $\displaystyle =a[x^4-(x_1+x_2+x_3+x_4)x^3+(x_1x_2+x_1x_3+x_1x_4+x_2x_3+ x_2x_4+x_3x_4)x^2-$

    $\displaystyle -(x_1x_2x_3+x_1x_2x_4+x_1x_3x_4+x_2x_3x_4)x+x_1x_2x _3x_4]=$

    $\displaystyle =a(x^4-10x^2+1)$
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