Construct a polynomial function with the following properties: third degree, only real coefficients, −3 and 5+i are two of the zeros, y-intercept is −78

Answer 1

imaginary zeros come in conjugate pairs

if 5 + i is one zero, then another zero is 5 -i

x= 5+i

x-5 = i

x^2 - 10x + 25 = -1

x^2 -10x +26 = 0

if 5-i is another zero, then

x = 5- i

x-5 = -i

x^2 -10x + 25 = (-i)^2 = -1

x^2 -10x +26 = 0 which is the same factor = (5

if 3 is a zero, then (x-3) is a factor, multiply the factors together

(x-3)(x^2 -10x +26)

= x^3 -13x^2 -56x - 78 is the 3rd degree polynomial with the given zeros, with only real coefficients and y intercept = -78