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Find a monic quartic polynomial f(x) with rational coefficients whose roots include x=5-i\sqrt[4]{3} . Give your answer in expanded form.

 Apr 26, 2021
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If \(5-i\sqrt[4]{3}\) is a root, then we can set \(5+i\sqrt[4]{3}\) to also be a root, since it is the conjugate of the other root and will multiply to cancel out the imaginary term. It would look something like this:

\((x-(5-i\sqrt[4]{3}))(x-(5+i\sqrt[4]{3}))\\=x^2-10x+25+\sqrt{3}\\=(x-5)^2+\sqrt{3}\)

Again, we could use the same conjugate trick as before to cancel out the radical:

\(((x-5)^2+\sqrt{3})((x-5)^2-\sqrt{3})\\=(x-5)^4-3\\=x^4 - 20 x^3 + 150 x^2 - 500 x + 625-3\\ =\boxed{x^4 - 20 x^3 + 150 x^2 - 500 x + 622}\)

(note that for the second last step you could use the binomial theorem to expand the terms faster.

 Apr 26, 2021

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