Polynomial function

Question

Find a polynomial fiction of least degree having only real coefficients, a leading coefficient of 1, and zeros of 1-I and 4-4i

Answer 1

zeros are 1-i and 4-4i

x=1-i

x-1 = -i

x^2 -2x+1 = -1

x^2 -2x+2 = 0

x^2 -2x +2 is one integer factor of the polynomial

x = 4-4i

x-4 = -4i

x^2 -8x +16 = -16

x^2 08x +32 is another factor

x=1

x-1 is the third factor

multiply them all together to get the 5th degree polynomial

f(x) = (x-1)(x^2-8x+32)(x^2 -2x +2)

=(x-1)(x^4 -8x^3 +34x^2 -16x +64)

= x^5 -9x^4 +42x^3 -48x^2 +80x -64= the Polynomial function with the given zeros

a cubic is the least degree polynomial with those 3 zeros. There are 2 additional zeros, as imaginary zeros come in conjugate pairs

Answer 2

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