p is a degree 5 polynomial that passes through the origin, has zeros i and 5 − i , and p ( 2 ) = 300 . Find the equation for p .

Question

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

Answer 1

Since it passes through the origin, 0 is also a zero. If the coefficients are real, the complex conjugate of any complex zero must also be a zero. Therefore your 5 zeros are

0

i

-i

5 - i

5 + i

p(x) = Ax (x - i) (x + i) [x - (5 - i)] [x - (5 + i)] = Ax (x² + 1) [x - (5 - i)] [x - (5 + i)]

Expand the expression on the right. The coefficients will come out real. Then set

p(2) = 300

and solve for A. You should be able to finish from here.