q is a degree 5 polynomial that passes through the origin, has zeros i and 2 − i , and q ( 1 ) = 12 . Find the equation for q .

Question

 is a degree 5 polynomial that passes through the origin, has zeros  and , and . Find the equation for .

Answer 1

Assuming q has real coefficients, the complex conjugates of complex zeros must also be zeros. Therefore the 5 zeros are

0 (given the curve passes through the origin)

i

-i

2 - i

2 + i

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

with A real and A ≠ 0. Expand this, and set

q(1) = 12

to find A. The coefficients should all come out real. You should be able to finish from here.