# Just Curious

As we all know, if $a_0+a_1x+...+a_nx^n=0$ has a solution of $p+qi$ where $\left\{ a_0,a_1,...,a_n \right\} \subset \mathbb{R}$/extract_itex], then $p-qi$ is also a solution. Similarly, if $a_0+a_1x+...+a_nx^n=0$ has a solution of $p+q\sqrt { m }$ where $\left\{ a_{ 0 },a_{ 1 },...,a_{ n } \right\} \subset \mathbb{Q}\ \quad and\quad \sqrt { m } \in \mathbb{I}\$, then $p-q\sqrt { m }$ is also a solution. I was playing around with it until I suddenly got curious. If $a_{ 0 }+a_{ 1 }x+...+a_{ n }x^{ n }=0\quad (n\ge 4\quad and\quad n\in \mathbb{N}\ )$ has a solution of $p\sqrt { m } +qi\quad (\sqrt { m } \in \mathbb{I}\ )$, then is it safe to assume that it also has the solutions of $p\sqrt { m } -qi,\quad -p\sqrt { m } +qi,\quad -p\sqrt { m } -qi\quad$? I tried it for $n=4$, and it worked. Can someone confirm this for all values of n's? Note by Nick Lee 6 years ago This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science. When posting on Brilliant: • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused . • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone. • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge. • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events. MarkdownAppears as *italics* or _italics_ italics **bold** or __bold__ bold - bulleted- list • bulleted • list 1. numbered2. list 1. numbered 2. list Note: you must add a full line of space before and after lists for them to show up correctly paragraph 1paragraph 2 paragraph 1 paragraph 2 [example link](https://brilliant.org)example link > This is a quote This is a quote  # I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world" # I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world" MathAppears as Remember to wrap math in $$ ... $$ or \[ ... $ to ensure proper formatting.
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Yes, your observation is correct. Can you figure out how to prove it?

Staff - 6 years ago

Let $p + q\sqrt{m}$ be a root of a polynomial $f(x)$.

We can write $f(x)$ = $(x-(p + q\sqrt{m})$$(x-(p - q\sqrt{m}))$$a(x)$ + $r(x)$

$r(x)$'s degree would be 1,

$\implies r(x) = ax + b$

If we get $a=b=0$,then $p - q\sqrt{m}$ will also be root of $f(x)$

Put $x = p + q\sqrt{m}$ & get $a=b=0$.

(This is not a proof but a hint.)

- 6 years ago

대입해서 복소수상등 무리수상등을 동시해하시면되는데 그게 힘들죠... 증명해보고잇어요!

- 5 years, 11 months ago