+647 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.
+2446 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?
+129852
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* x3 * 3 + 6x2 * 32 - 4x * 33 + 34 - 2 =
x^4 - 12x^3 + 54x^2 - 108x + 79 = P(x)
