How do you write a polynomial with zeros 4-i and sqrt(10)?

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Answer 1 · 2016-08-16T20:32:18

This one works:

$x^2-(4+sqrt(10)-i)x+(4sqrt(10)-(sqrt(10))i)$

...but you probably want this one:

$x^4-8x^3+7x^2+80x-170$

If you allow coefficients of arbitrary type, then the polynomial of lowest degree with these zeros is simply:

$(x-(4-i))(x-sqrt(10))$

$= x^2-(4+sqrt(10)-i)x+(4sqrt(10)-(sqrt(10))i)$

If you want rational coefficients, then include the Complex conjugate $4+i$ and the radical conjugate $-sqrt(10)$ as two more zeros to find:

$(x-(4-i))(x-(4+i))(x-sqrt(10))(x+sqrt(10))$

$=((x-4)^2-i^2)(x^2-10)$

$=(x^2-8x+17)(x^2-10)$

$=x^4-8x^3+7x^2+80x-170$