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.

waffles Jan 4, 2018

#1**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?

TheXSquaredFactor Jan 4, 2018

#2**+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 }* 3 + 6x^{2 }* 3^{2} - 4x * 3^{3} + 3^{4} - 2 =

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

CPhill Jan 6, 2018