Question

Find the polynomial with real coefficients of the smallest possible degree for which 4i and 1+i are zeros and in which the coefficient of the highest power is 1.

Answer 1

Complex conjugate theorem states that for a polynomial with real coefficients , complex zeros occur in conjugate pair. That is if a+ib is zero of a polynomial with real coefficients then a-ib is also zero of that polynomial .

Now in our case if 4i and 1+i are zero of a polynomial with real coefficients then -4i and 1-i are also zero of that polynomial. So (x-4i), (x-(-4i)), (x-(1+i)) and (x-(1-i)) are the required polynomial.

Also we need a polynomial with coefficient of highest power is 1.

So the required polynomial is

(x-4i)(x+4i)(x-1-i)(x-1+i)

=(x^2+16)((x-1)^2+1) { using (a+b)(a-b)=a^2+b^2 and i^2=-1}

=(x^2+16)(x^2-2x+2)

=x^4-2x^3+18x^3-32x+32

So this polynomial is of lowest degree which satisfies the given conditions.