Write a polynomial function of minimum degree in standard form with real coefficients whose zeros and their multiplicities include those listed. 6 ​(multiplicity 2), 2+i ​(multiplicity 1)

Question

Write a polynomial function of minimum degree in standard form with real coefficients whose zeros and their multiplicities include those listed. 6 ​(multiplicity 2), 2+i ​(multiplicity 1)

Answer 1

x=2+i

x-2=i

x^2-4x+4=-1

x^2-4x+5=0

x=6

x-6=0

x^2-12x+36=0

f(x) = (x^2-4x+5)(x^2-12x+36)

f(x) = x^4 -16x^3 +89x^2 -204x +180

there are 4 zeros, as imaginary zeros come in conjugate pairs

if 2+i is a zero, so is 2-i

4 zeros mean 4th degree polynomial