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Math Help - Polynomial cubic roots

  1. #1
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    Polynomial cubic roots

    If the roots of the equation xcubed + 3xsquared - 2x +1 = 0 are e, f, g
    find the value of
    a) esquared(f+g) + fsquared(g+e) + gsquared(e+f)
    b) esquaredfsquared + fsquaredgsquared + gquaredesquared

    this knowing that e+f+g=-b/a, ef+eg+fg=c/a and efg=-d/a (general cubic equation rules)

    AND

    Solve the equation 4xcubed - 12xsquared +9x - 2 =0 given that two of its roots are equal

    thankyou in advance for any help given ! appreciate it !
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  2. #2
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    Quote Originally Posted by iiharthero View Post
    Solve the equation 4xcubed - 12xsquared +9x - 2 =0 given that two of its roots are equal
    4x^3 - 12x^2 + 9x - 2 = 0

    By the rational roots theorem, the possible rational roots are
    \frac{\text{factors of}\; -2}{\text{factors of}\; 4} = \frac{\pm 1, \pm 2}{\pm 1, \pm 2, \pm 4}
    = 1, 2, 1/2, 1/4.

    Start testing these factors by using the Factor Theorem or synthetic division. You'll find that x = 2 is a root:
    Code:
    2| 4 -12   9  -2
    --     8  -8   2
    -----------------
       4  -4   1   0
    So the polynomial factors into
    \begin{aligned}<br />
4x^3 - 12x^2 + 9x - 2 &= (x - 2)(4x^2 - 4x + 1) \\<br />
&= (x - 2)(2x - 1)(2x - 1)<br />
\end{aligned}

    The other roots are x = 1/2 (double root).


    01
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  3. #3
    Senior Member Stroodle's Avatar
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    Are you sure that x^3+3x^2-2x+1=0 is the right equation? Because it only has one root.
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  4. #4
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    Hello, iiharthero!

    Here's part (b) . . .


    If the roots of: . x^3 + 3x^2 - 2x +1 \:=\: 0 . are e, f, g

    find the value of:

    (a)\;\;e^2(f+g) + f^2(g+e) + g^2(e+f)

    (b)\;\;e^2f^2 + f^2g^2 + g^2e^2
    (b) .We have: . \begin{array}{ccc}<br />
e+f+g \:=\:\text{-}3 & {\color{blue}[1]} \\ ef+fg+ge\:=\: \text{-}2 & {\color{blue}[2]} \\ efg \:=\: 1 & {\color{blue}[3]} \end{array}


    Square [2]: . (ef+fg+ge)^2 \:=\:(\text{-}2)^2

    . . e^2f^2 + 2ef^2g + 2e^2fg + f^2g^2 + 2fg^2e + g^2e^2 \:=\:4

    . . (e^2f^2+f^2g^2+g^2e^2) + 2\underbrace{efg}_{1}\underbrace{(e+f+g)}_{-3} \:=\:4

    . . (e^2f^2 + f^2g^2 + g^2e^2) + 2(1)(-3) \:=\:4


    Therefore: . e^2f^2 + f^2g^2 + g^2e^2 \:=\:10

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