How do you write a polynomial function of least degree and leading coefficient 1 when the zeros are 3-i, 5i?

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Answer 1 · 2016-11-27T07:02:11

$x^{4} - 6 x^{3} + 35 x^{2} - 150 x + 250$ $x^4-6x^3+35x^2-150x+250$ .

Complex roots occur in conjugate pairs. The zeros of the required

polynomial are $\pm 5 i and 3 \pm i$ $+-5i and 3+-i$ m and so it is

$((x - 5 i) (x + 5 i)) ((x - 3) - i) ((x - 3) + i))$ $((x-5i)(x+5i))((x-3)-i)((x-3)+i))$

$= (x^{2} + 25) ({(x - 3)}^{2} + 1)$ $=(x^2+25)((x-3)^2+1)$

$= (x^{2} + 25) (x^{2} - 6 x + 10)$ $=(x^2+25)(x^2-6x+10)$

$= x^{4} - 6 x^{3} + 35 x^{2} - 150 x + 250$ $=x^4-6x^3+35x^2-150x+250$ .

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