How do you write a polynomial function of least degree with integral coefficients that has the given zeroes 6, 2i?

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How do you write a polynomial function of least degree with integral coefficients that has the given zeroes 6, 2i?

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Answer 1 · 2016-07-17T07:29:12

$f (x) = x^{3} - 6 x^{2} + 4 x - 24$ $f(x) = x^3-6x^2+4x-24$

Explanation:

If a polynomial has Real coefficients then any non-Real zeros will occur in Complex conjugate pairs. So if $2 i$ $2i$ is a zero, then so is $- 2 i$ $-2i$ .

Hence our polynomial function can be written:

$f (x) = (x - 6) (x - 2 i) (x + 2 i)$ $f(x) = (x-6)(x-2i)(x+2i)$

$= (x - 6) (x^{2} + 4)$ $= (x-6)(x^2+4)$

$= x^{3} - 6 x^{2} + 4 x - 24$ $= x^3-6x^2+4x-24$

Any polynomial in $x$ $x$ with these zeros will be a multiple (scalar or polynomial) of this $f (x)$ $f(x)$

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How do you write a polynomial function of least degree with integral coefficients that has the given zeroes 6, 2i?

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