# 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

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

Would I use some kind of factoring of $$(x+y)^4$$ I am guessing that there will a step to get rid of the imaginary value, and then a step to get rid of the irrational values, but I do not know how to approach this... Can someone please give me a hint?

Thanks!

• what is $(x-3)^4 \; ? \;$ Feb 21 '20 at 19:58
• Use conjugates; e.g., multiply $3-i\sqrt[4]2$ by $3+i\sqrt[4]2$; but Will Jagy’s hint is better Feb 21 '20 at 20:00
• $(x-3)^4=x^4+12x^3+54x^2+108x+81$ How does this help us...? Feb 21 '20 at 23:23

\begin{aligned} x&= 3 - i \sqrt[4]{2} \\ x-3&= -i \sqrt[4]{2} \\ (x-3)^4&= 2 \\ (x-3)^4-2&=0 \\ x^4 - 12 x^3 + 54 x^2 - 108 x + 79&=0 \end{aligned}
EDIT. In fact, you can even factor the given polynomial fully (if you desired) because examining the construction closely you can see the roots of $$x^4 - 12 x^3 + 54 x^2 - 108 x + 79$$ are precisely, $$3 \pm i \sqrt[4]{2}$$ and $$3 \pm \sqrt[4]{2}$$, i.e. $$3 + \sqrt[4]{2} e^{2\pi ki/4}$$ for $$k=0,1,2,3$$.