Question

Write an equation in general form for a polynomial function with real coefficients and zeros at 1 and 3 + 5i.

Answer 1

to have real coefficients, any complex roots come in complex conjugate pairs.
if you have a+ bi then you must also have a - bi (complex conjugate means the imaginary part is negated)

so you will need three roots for this problem
multiply out (x- a) (x-b) (x-c)
where a, b and c are the roots.

first step: what are the 3 roots ?

Answer 2

I,m not 100% sure, but I believe the roots are (x-1) (x+3) (x+5i)

Answer 3

x=1 is a root
subtract 1 from both sides: x-1= 0
that means (x-1) is one of the factors

x= 3 + 5i
so (x- (3+5i))=0 or (x-3-5i) =0 and (x-3-5i) is another factor

you need the complex conjugate of 3+5i for the 3rd root. It is 3-5i
x= 3-5i
the 3rd factor is (x-3+5i)

now you have to multiply out
(x-1)(x-3-5i)(x-3+5i)

and remember that i*i = -1