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Math Help - Roots of polynomials

  1. #1
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    Smile Roots of polynomials

    Hi, sorry if this is in the wrong forums not really sure but here it goes..

    The quadratic equation 2x^2+4x+3=0 has roots \alpha and \beta .

    Find the value \alpha^4+\beta^4

    Ive managed to expand to \alpha^4+\beta^4=(\alpha+\beta)^4-4\alpha^3\beta-6\alpha^2\beta^2-4\alpha\beta^3

    but im not sure how to put in terms of \alpha\beta and \alpha+\beta to work out an answer.

    Also the previous part of the question asks to show \alpha^2+\beta^2=1 which I managed to prove, is there anyway I can use this to answer the question?

    Help really appreciated. Thankyou !
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  2. #2
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    Hello, NathanBUK!

    The quadratic equation 2x^2+4x+3\:=\:0 has roots \alpha and \beta .

    Find the value of: \alpha^4+\beta^4


    Also the previous part of the question asks to show \alpha^2+\beta^2=1
    . . which I managed to prove. . . . . Good!

    Is there anyway I can use this to answer the question? . . . . Yes!

    We know that: . \begin{Bmatrix}\alpha + \beta &=& -2 \\ \alpha\beta &=& \frac{3}{2} \end{Bmatrix}

    We have: . \alpha^2 + \beta^2 \;=\;1

    Square: . \left(\alpha^2 + \beta^2\right)^2 \;=\;(1)^2

    . . . . \alpha^4 + 2\alpha^2\beta^2 + \beta^4 \;=\;1

    . . . \alpha^4 + \beta^4 + 2\!\cdot\!\!\!\underbrace{(\alpha\beta)^2}_{\text{  This is }(\frac{3}{2})^2} \;=\;1

    . . . . . \alpha^4 + \beta^4 + \frac{9}{2} \;=\;1

    . . . . . . . \alpha^4 + \beta^4 \;=\;-\frac{7}{2}

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  3. #3
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    Ohh thats makes soo much sense now!! Thankyou so much!!
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by NathanBUK View Post
    Hi, sorry if this is in the wrong forums not really sure but here it goes..

    The quadratic equation 2x^2+4x+3=0 has roots \alpha and \beta .

    Find the value \alpha^4+\beta^4

    Ive managed to expand to \alpha^4+\beta^4=(\alpha+\beta)^4-4\alpha^3\beta-6\alpha^2\beta^2-4\alpha\beta^3

    but im not sure how to put in terms of \alpha\beta and \alpha+\beta to work out an answer.

    Also the previous part of the question asks to show \alpha^2+\beta^2=1 which I managed to prove, is there anyway I can use this to answer the question?

    Help really appreciated. Thankyou !
    Alternative solution (same one in disguise realy):

    \alpha^4+\beta^4=(\alpha^2+\beta^2)^2-2(\alpha \beta)^2

    CB
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