Find a polynomial f(x) of degree 4 with real coefficients and the following zeros.

Question

2, -4, 2-i

Answer 1

zeros 2, -4, 2-i, 2+i (imaginary zeros come in conjugate pairs)

corresponding factors are (x-2)(x+4)(x^2-4x+2)

x = 2-i

x-2 = -i

x^2 -4x +1 = -1

x^2 -4x +2 = 0

multiply the above 3 factors together to get a 4th degree polynomial with the given 3 zeros and implicit 4 zeros

f(x) = (x^2+2x-8)(x^2-4x+2)

f(x) = x^4 -2x^3 -14x^2 +36x -16

check the real zeros, plug in 2 and -4 for x

f(2) = 16-16 -56 + 72-16

=0

f(-4) = 4^4 +128 -14(16) - 144 -16

= 256 +128 - 240-144

=0