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The polynomial equation \[x^3 + bx + c = 0,\]where $b$ and $c$ are rational numbers, has $3-\sqrt{7}$ as a root. It also has an integer root. What is it?

 

Why I'm confused, so in my last repost, melody, a great answered wanted me to find the root with the conjugate, with the factor theorem, 

its true you are allowed to do that if the coefficients are real. I obviously did that and I got $x^2-6x+2$. Do I use synthetic division and divide it? I'm really confused since I know synthetic division but I don't know with variables...

 
 Jan 12, 2021
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Yes....just do regular longhand division....

     for no remainder to be true , you will find   b-2 = -36     (so b = -34)      and c = 12      

       and the other root is x = -6

 
 Jan 12, 2021

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