Find a polynomial f(x) of degree 3 with real coefficients and the following zeros.

Question

-3, 2+3i

Answer 1

-3 and 2+3i are 2 factors

imaginary zeros come in congugate pairs

if 2+3i is a zero, 2-3i is a 3rd zero

the 3 degree polynomial with real coefficients is

(x+3)(x^2-4x+13) = x^3 -x^2 +x+69

factors are the x zeros with changed signs, with an x stuck in front of them

then multiply them together

x- 3= 0, x=3

x-2+3i = 0

x = 2-3i

x= 2+3i

x=2+/-3i

x-2 = +/-3i

(x-2)^2 = -9

x^2-4x+4 =-9

x^2-4+13 =0

x^2-4x+13 is one factor with real coefficients

the other factor is x+3

multiply them together to get

x^3-x^2+x+69, the 3 Degree Polynomial