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OpenStudy (anonymous):

0, 0, 4, 1+i find a polynomial function with integer coefficients that has the given zeroes

OpenStudy (asnaseer):

I a polynomial has roots \(x=r_1,x=r_2,x=r_3,...,x=r_n\) then it can be written as follows:\[(x-r_1)(x-r_2)(x-r_3)...(x-r_n)=0\]so, for the roots you have provided, we can write the polynomial as:\[\begin{align} (x-0)(x-0)(x-4)(x-(1-i))&=0\\ \therefore (x)(x)(x-4)(x-1+i)&=0\\ \therefore x^2(x-4)(x-1+i)&=0\\ \therefore x^2(x^2-x+ix-4x+4-4i)&=0\\ \therefore x^2(x^2-5x+ix+4-4i)&=0\\ \therefore x^2(x^2-(5-i)x+4(1-i))&=0\\ \therefore x^4-(5-i)x^3+4(1-i)x^2&=0\\ \end{align}\]

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