Question

write a polynomial function with real coefficients in standard form whose zeros include -2, 3 and 4-5i

Answer 1

\[(x+2)(x-3)(x-(4+5i))(x-(4-5i))\] is start

Answer 2

multiplying the first two terms is easy, you get
\[(x+2)(x-3)=x^2-x-6\]

Answer 3

it is the second two that is somewhat annoying, but you have a couple says to do it

Answer 4

one way is to work backwards
if a zero is \(4+5i\) then you can write
\[x=4+5i\]
\[x-4=5i\]
\[(x-4)^2=-25\]
\[x^2-8x+16=-25\]
\[x^2-8x+41=0\] is your polynomial

Answer 5

another way is to remember that if \(a+bi\) is the zero of a quadratic polynomial, the quadratic is
\[x^2-2ax+a^2+b^2\] and in your case \(a=4,b=5\)