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Math Help - REal Zeros

  1. #1
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    REal Zeros

    Find all the real zeros of the polynomial f(x)= X^4-80x^2+1024 and determine the multiplicity of each.
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  2. #2
    MHF Contributor Amer's Avatar
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    Quote Originally Posted by refresh View Post
    Find all the real zeros of the polynomial f(x)= X^4-80x^2+1024 and determine the multiplicity of each.
    let

    u=x^2

    f(u) = u^2 - 80u +1024


    u = \frac{-(-80) \mp \sqrt{(80)^2 -4(1024)}}{2}

    u = \frac{80 \mp 48}{2}

    u = 40+24 = 64

    u = 40-24 = 16

    so

    f(u) = (u-16)(u-64) but

    f(x) = (x^2-16)(x^2-64)

    f(x)=(x-4)(x+4)(x-8)(x+8)

    roots

    x={-8,-4,4,8}

    the multiplicity is one for all roots
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  3. #3
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    Quote Originally Posted by Amer View Post
    let

    u=x^2

    f(u) = u^2 - 80u +1024


    u = \frac{-(-80) \mp \sqrt{(80)^2 -4(1024)}}{2}

    u = \frac{80 \mp 48}{2}

    u = 40+24 = 64

    u = 40-24 = 16

    so

    f(u) = (u-16)(u-64) but

    f(x) = (x^2-16)(x^2-64)

    f(x)=(x-4)(x+4)(x-8)(x+8)

    roots

    x={-8,-4,4,8}

    the multiplicity is one for all roots
    OMG!!! Thank you so much. i have been waiting for help on this for 2 hours you are the best! i was stuck because i didnt know what multiplicity was. so i got confuse ass to is the 2 multipicity or 1 because its a positive and a negative. well thank you so much.
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  4. #4
    MHF Contributor Amer's Avatar
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    Quote Originally Posted by refresh View Post
    OMG!!! Thank you so much. i have been waiting for help on this for 2 hours you are the best! i was stuck because i didnt know what multiplicity was. so i got confuse ass to is the 2 multipicity or 1 because its a positive and a negative. well thank you so much.
    multiplicity is the power of the roots

    example suppose we have this function

    f(x)= (x-3)^2 (x+2)^4(x-1)

    here the root -2 has the multiplicity 4
    the root 3 has the multiplicity 2
    the root 1 has the multiplicity 1
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