We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive pseudonymised information about your use of our website. cookie policy and privacy policy.
 
+0  
 
0
1103
2
avatar+644 

Find a monic quartic polynomial f(x) with rational coefficients whose roots include x=3-i\sqrt[4]2 . Give your answer in expanded form.

 Jan 4, 2018
 #1
avatar+2338 
0

waffles, I see that the phrase "whose roots include x=3-i\sqrt[4]2" has a typo, I presume. Can you clarify if the 2 is included inside of the square root, whether it should be omitted, or whether it is something else entirely?

 Jan 4, 2018
 #2
avatar+100529 
+1

 

 

If   3 -  4√2 i    is a root, then so is the conjugate   3 +  4√2 i

 

Let  the polynomial we are looking for be P(x)

 

And  suppose that P(x)  can be written as  Q(x) * R(x)

 

So ....let Q(x) be  formed by

 

( x -  ( 3 - 4√2 i) )  ( x -  (3  + 4√2 i) )   =

 

[ ( (x - 3) +  4√2 i  ]    [  (x - 3)  -  4√2 i  ]   =

 

(x - 3)2   +  √2

 

And it's clear that we will have a polynomial with rational coefficients if  R(x)  is

 

(x - 3)2   - √2

 

So

 

      Q(x)            *         R(x)        

 

[( x - 3)2  + √2 ] *  [( x - 3)2  - √2 ]    produces

 

(x - 3)4   -  2    =

 

x^4  -  4* x* 3  + 6x* 32  -  4x * 33  + 34  -  2   =

 

x^4  -  12x^3  + 54x^2  - 108x  +  79   =  P(x)

 

 

cool cool cool

 Jan 6, 2018
edited by CPhill  Jan 6, 2018
edited by CPhill  Jan 6, 2018

21 Online Users

avatar