Find the polynomial with complex coefficients of the smallest possible degree for which I and 1+I are zeros and in which the coefficient of the highest power is 1.

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Marian Tucker · Accepted Answer

To find the polynomial with complex coefficients  Suppose f(x) be the polynomial with complex coefficients whose zeroes are i and (1+i)  Since the zeroes i and (1+i) are complex number and so its conjugates are also the zeroes of the polynomial f(x) with the real coefficient.  If  a is a zero of the polynomial f(x), then (x-a) is a factor of the polynomial f(x).  So, (x-i) and [x-(1+i)] are factors of the polynomial f(x).  Then, the polynomial f(x) will be of degree 2.  So,  f(x)=(x-i)[x-(1+i)]  =x2−x−ix−ix+i+i2  =x2−x−2ξ+i−1  =x2−(1+2i)x+(i−1)  Hence, the required polynomial with the complex coefficient of the smallest degree is  f(x)=x2−(1+2i)x+(i−1)

Find the polynomial with complex coefficients of the smallest possible

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