f is a degree 5 polynomial that passes through the origin, has zeros i and 2 − i, and f ( − 4 ) = − 12580. Find the equation for f.

Question

answer has to be simplified so that it only has real numbers

Answer 1

I will show you how to solve this, but YOU do the work. The complex conjugate of a complex zero must also be a zero, so your 5 zeros are

0

i

-i

2 - i

2 + i

f(x) = Ax (x - i)(x + i) [x - (2 - i)] [x - (2 + i)]

f(x) = Ax (x² + 1) [x - (2 - i)] [x - (2 + i)]

Now carefully expand this and simplify. All the coefficients will come out real. Then set

f(-4) = -12580

and solve for A, which will also come out real. You should be able to finish from here.