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find a monic polynomial of lowest possible degree with rational coefficeients and having both 2 and 3-i roots.

 Feb 17, 2021
 #1
avatar+120287 
+3

If  3  - i   is a root....so is  3 + i

 

So  the polynomial  is of the  form

 

x^3  +  bx^2  + cx  +  d

 

b =  - sum of  the roots  =   =  -[ 2 + (3 + i)  + (3  - i)]  =  - 8

 

c =  sum of the product of  roots  taken two at a time  =    2(3 + i)  + 2 (3 -i)  + ( 3 + i) (3  -i)  =   6  + 6  + 9  - i^2  =  12  + 9 + 1   =   22

 

d  =  -  product of  the roots  = - 2 (3+i) ( 3 - i)   =  -  2 ( 9 -i*2)   = - 2 ( 9 + 1  )    =  -20

 

The polynomial  is

 

x^3     - 8x^2   +  22x   -  20

 

 

cool cool cool

 Feb 17, 2021
edited by CPhill  Feb 17, 2021
 #2
avatar+108 
+2

Oh I didn't think about using the conjugate. Thanks!

whatdoiputhere  Feb 17, 2021

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