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Math Help - determine values of ω, that satisfy the roots given.....

  1. #1
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    Question determine values of ω, that satisfy the roots given.....

    i have an assignment question hopefully someone can help me with or point me in the right direction....

    the question comes in 3 parts.i have done parts (i),(ii) i need to work out the 3rd part(iii).

    (ii) if the roots of a cubic equation are x= -5/2 , x= 1/4, x= 3
    determine the cubic expression by expansion, with these roots.


    (iii) if x = Cos(2ω) in part (ii) of this question where ω is measured in radians. determine the values of ω, that satisfy the roots given that
    0 ≤ ω ≥ 2(pi)


    any help much appreciated

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  2. #2
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    Hello, decoy808!

    The question comes in 3 parts. I have done parts (i), (ii).

    (ii) If the roots of a cubic equation are: x\:=\:\text{-}\tfrac{5}{2},\:\tfrac{1}{4},\:3
    determine the cubic expression, by expansion, with these roots.
    We have: . \begin{array}{c}x \:=\:\text{-}\frac{5}{2} \\ \\[-4mm] x \:=\:\frac{1}{4} \\ \\[-4mm] x \:=\:3 \end{array}\quad\Rightarrow\quad \begin{array}{cc} 2x+5 \:=\:0 &{\color{blue}[1]}\\ 4x-1 \:=\:0 &{\color{blue}[2]}\\ x-3\:=\:0 & {\color{blue}[3]} \end{array}<br />

    Multiply [1]. [2] and [3]: . f(x) \;=\;(2x+5)(4x-1)(x-3)


    Therefore: . f(x) \;=\;8x^3 - 6x^2 - 59x + 15




    (iii) x = \cos2\theta in part (ii) of this question, where \theta is measured in radians.
    Determine the values of \theta that satisfy the roots, given that  0 \leq \theta \leq 2\pi
    We have: . \left(2\cos2\theta + 5\right)\left(\cos2\theta - 3\right)\left(4\cos2\theta - 1\right) \:=\:0


    And we have three equations to solve:

    . . 2\cos\theta + 5 \:=\:0\quad\Rightarrow\quad \cos2\theta \:=\:\text{-}\tfrac{5}{2} \quad\hdots\quad\text{ No real roots.}

    . . \cos2\theta - 3 \:=\:0 \quad\Rightarrow\quad \cos2\theta \:=\:3 \quad\hdots\quad\text{ No real roots.}


    . . 4\cos2\theta - 1 \:=\:0 \quad\Rightarrow\quad \cos2\theta \:=\:\tfrac{1}{4}

    . . . . 2\theta \;\approx\;\begin{Bmatrix}1.318116 \\ 4.965069 \\ 7.601301 \\ 11.248255 \end{Bmatrix} \quad\Rightarrow\quad \theta \;\approx\;\begin{Bmatrix}0.654 \\ 2.483 \\ 3.801 \\ 5.624 \end{Bmatrix}

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  3. #3
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    thanks soroban it is a great help. i am still trying to understand a few things please excuse my ignorance

    why did u subsitute (omega)for (theta)?
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  4. #4
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    Hello, decoy808!

    why did u subsitute (theta)for (omega)?
    I's just easier to type.

    My fingers are used to typing \-t-h-e-t-a,
    . . while I have spell \-o-m-e-g-a very carefully.


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