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Can you find a quartic (degree 4) polynomial $p(x) = ax^4+bx^3+cx^2+dx+e$ with real coefficients $a$, $b$, $c$, $d$, $e$ whose roots are precisely $x=5$, $x=-2$, $x=3$ and $x=1+i$ ?

Guys please help I've been pondering this question for hours and still can't find it :S

Thanks a lot!

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what do you know about the complex roots of a polynomial with real coefficients? – N. S. Oct 12 '12 at 4:04
Can you give me any hint? :S – John Lee Oct 12 '12 at 4:07
up vote 1 down vote accepted

No. With real coefficients, if some complex number $\beta$ is a root, then so is $\bar{\beta}$

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No, you can't, since the product of the roots is a complex non-real number, which equals the polynomial's free coefficient divided by the main one:

$$(-2)\cdot 3\cdot 5\cdot(1+i)=30+i\in\Bbb C\setminus \Bbb R$$

If $\,a\,,\,e\in\Bbb R\,$, then also $\,e/a\in\Bbb R\,$

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