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Math Help - Two exercises - polynomial equations and geometric and arithmetic sequences

  1. #1
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    Exclamation Two exercises - polynomial equations and geometric and arithmetic sequences

    Hey, my neighbour again send me one exercise.


    1. In the equation x^{3} - 7x^{2} - 21x + a = 0 you must find solutions whose are in geometric sequence. For which a?


    How can I find this a?


    2. In the equation x^{4} -(a+3)x^2 + (a + 2) = 0 must find this a that solutions are in arithmetic sequence.


    How can I find this a?
    Last edited by lebdim; August 12th 2014 at 01:38 PM.
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  2. #2
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    Re: Two exercises - polynomial equations and geometric and arithmetic sequences

    (1) A cubic equation can have up to 3 distinct solutions. If they are in geometric progression, we can write them as u, ur, and ur^2. If those are the solutions to a cubic equation with this equation, you must have (x- u)(x- ur)(x- ur^2)= x^3- 7x^2- 21x+ a. Multiply out on right and compare the coefficients of x^2 and x on left and right to get two equations for u and r. Once you have those it is easy to find a.

    (2) Pretty much the same thing. A quartic equation can have up to 4 distinct solutions. If the solutions are in arithmetic progression, we can write them as u, u+ r, u+ 2r, and u+ 3r. So we must have (x- u)(x- u- r)(x- u- 2r)(u+ 3r)= x^4- (a+ 3)r^2+ (a+ 2).
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  3. #3
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    Re: Two exercises - polynomial equations and geometric and arithmetic sequences

    Hello, lebdim!

    1. In the equation x^3 - 7x^2 - 21x + b = 0, the solutions are in geometric sequence.
    . . For which b?

    The roots are: . a,\,ar,\,ar^2.

    Using Vieta's formulas, we have:

    . . \begin{Bmatrix}a+ar+ar^2 \:=\:7 & \Rightarrow & a(1+r+r^2) \:=\:7 & [1] \\ a^2r + a^2r^2 + a^2r^3 \:=\:-21 & \Rightarrow& a^2r(1+r+r^2) \:=\:-21 & [2] \\ a^3r^3 \:=\:-b & \Rightarrow & (ar)^3 \:=\:-b & [3] \end{Bmatrix}

    Divide [2] by [1]: . \frac{a^2r(1+r+r^2)}{a(1+r+r^2)} \:=\:\frac{-21}{7} \quad\Rightarrow\quad ar \:=\:-3

    Cube both sides: . (ar)^3 \:=\:(-3)^3 \:=\:-27

    Substitute into [3]: . -b \:=\:-27 \quad\Rightarrow\quad \boxed{b \:=\:27}
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