If a polynomial function with rational coefficients has the zeros -5 and i, what are the additional zeros?

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Answer 1 · 2016-11-15T12:19:28

Additional zero is $-i$ and polynomial function is $x^3+5x^2+x+5$

A polynomial function whose zeros are $alpha$ , $beta$ , $gamma$ and $delta$ and multiplicities are $p$ , $q$ , $r$ and $s$ respectively is

$(x-alpha)^p(x-beta)^q(x-gamma)^r(x-delta)^s$

It is apparent that the highest degree of such a polynomial would be $p+q+r+s$ .

In the given case as two zeros are $-5$ and $i$ and coefficients are rational, assuming multiplicity of each to be just $1$ ,

we will have to have additionally a zero $-i$ (being complex conjugate of $i$ ) and hence polynomial is

$(x-(-5))(x-i)(x-(-i))$

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

= $(x+5)(x^2+1)$

= $x^3+5x^2+x+5$