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The roots of x^3-6x*^2-13x-k=0 form an artithmetic sequence.

a) What is the median term of the arithmetic sequence

b) Find the value of k

 Feb 17, 2021
 #1
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Call the roots   a , a + d   and  a  + 2d

 

Sum  of  the roots =  6

Sum of product of the  roots  taken two at a time  =  -13

 

So  we  have  this system

 

a + a + d  +  a  + 2d  =  6

a(a + d) +  a (a + 2d)  + (a + d) ( a + 2d)   =  -13

 

A little tedious to  solve this....so...I'm leaning on WolframAlpha  ( I can go through it if you  want me to)

 

We have  two possible  arithmetic  sequences

 

First one

a = -3   d  = 5

So

-3 , 2 , 7      are  the roots

So

( x + 3) ( x - 2) ( x - 7)    = x^3 - 6 x^2 - 13 x + 42  so k  =  -  42

 

 

Second one

a  = 7   d  =  -5

So   again

7 , 2 , -3     are  the roots and we  get  the same  value  for  k

 

And  the  median term    =  2

 

 

cool cool cool

 Feb 17, 2021

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