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# 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.

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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

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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
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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)

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