Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros. Write the polynomial in standard form. 2, 1+i, 2−3√

Answer 1

(x-2)(x^2-2x+2)(x^2-4x+1) =

x^5 -8x^4 + 23x^3 -32x^2 + 22x -4

there are 5 roots or zeroes. the imaginary & irrational roots come in conjugate pairs

1+i and 1-i, 1+sqr3 and 1- sqr3. so you end up with a 5th degree polynomial

x=1+i

x-1 = i

(x-1)^2 =-1

x^2-2x+1 =-1

x^2-2x+2 =0

x^2-2x+2 is a factor with 2 zeroes: 1+i and 1-i

x=1-sqr3

x-1 = -sqr3

(x-1)^2 =3

x^2 -2x +1 = 3

x^2 -2x -2 = 0

x^2 -2x -2 is a factor with 2 zeroes: 1-sqr3 and 1+sqr3

x=2

x-2 = 0

x-2 is a factor with 1 zero: 2