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(a) Prove that if the roots of \(x^3 + ax^2 + bx + c = 0\) form an arithmetic sequence, then \(2a^3 + 27c = 9ab.\)

 

(b)  Prove that if \(2a^3 + 27c = 9ab,\) then the roots of \(x^3 + ax^2 + bx + c = 0\) form an arithmetic squence. 

 

-------Thanks! coollaugh

 Apr 22, 2020
 #1
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a)  Let the first root be "r", then since the roots form an arithmetic sequence, the other two roots are "r + d" and "r + 2d".

     The factors will be:  (x - r)(x - (r + d))(x - (r + 2d))  or  (x - r)(x - r - d)(x - r - 2d)

 

     Expanding  (x - r)(x - r - d)(x - r - 2d)  =  x3 -3rx2 - 3dx2 + 3r2x + 6drx + 2d2x - r3 - 3dr2 - 2d2r

 

     which means that      a  =  -3r - 3d          b  =  3r2 + 6dr + 2d2          c  =  -r3 - 3dr2 - 2d2r

 

    2a3  =  -54r3 - 162r2d - 162rd2 - 54d3

    27c  =  -27r3 - 81dr2 - 54d2r

    9ab  =  -81r3 - 162r2d - 54d2r - 81r2d - 162d2r - 54d3

 

So, 2a3 + 27c  =  9ab

 

[It wasn't easy ...]

 Apr 22, 2020

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