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Math Help - Find the solutions of this polynomial

  1. #1
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    Find the solutions of this polynomial

    p(x) = 3x^4 - 11x^3 + 35x^2 + 21x - 26

    if one of the solutions is 2 + 3i.


    ok, so its conjugate is also a solution? 2 - 3i?

    i multiplied the conjugates and then applied long division

    and got 3x^2 + x + 2

    but im stuck..

    how do i get the other two solutions? is the above answers even right?

    thankz
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  2. #2
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    Quote Originally Posted by NeedHelp18 View Post
    p(x) = 3x^4 - 11x^3 + 35x^2 + 21x - 26

    if one of the solutions is 2 + 3i.
    Are you sure you have written the problem correctly? 2+3i is not a root of p.
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  3. #3
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    Hello, NeedHelp18!

    There seems to be a typo . . . the middle term is 33x^2, isn't it?


    p(x)\;=\;3x^4 - 11x^3 + {\color{red}33}x^2 + 21x - 26 \:=\:0 .if one of the roots is 2 + 3i.

    If 2 + 3i is a root, then so is 2 - 3i

    Then: [x - (2+3i)] and [x - (2-3i)] are factors.


    Dividing by: [x-(2i+3)][x-(2i-3)] \:=\:x^2 - 4x + 13, we get:

    . . p(x) \;=\;(x^2-4x+13)\,(3x^2+x-2)\:=\:0 \quad\Rightarrow\quad (x^2-4x+13)(x+1)(3x-2) \:=\:0


    Therefore, the other two roots are: . -1,\:\tfrac{2}{3}

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