Given 8i is a root of polynomial, P(x) = x^3 - 4x^2 + 64x - 256, find the remaining roots and write in factored form.

Question

Find a possible formula for P(x). You do not need to simplify your answer.

P(x) =

Answer format should look like (x-a)(x-bi)(x+bi))

Answer 1

8i and -8i are roots

imaginary roots always come in conjugate pairs

x=8i

x^2 =-64

x^2+64 is a factor

divide the 3D polynomial by that factor to find the other factor

set it equal to zero to solve for the other root

quotient is: x-4. Use long or synthetic division

P(x) = (x-4)(x^2+64)

or if you allow imaginary factors then

P(x) = (x-4)(x-8i)(x+8i)

most problems like this though

restrict the factors to integer or rational factors

if you go down that imaginary, irrational road,

(like the yellow brick road Dorothy & friends followed in Wizard of Oz)

you could also factor it even further

as

P(x) =(sqrx -2)(sqrx+2)(x-8i)(8+i)

but no one ever does that,

at least not that I've ever seen

then you could go even further

P(x) = (x^(1/4)-sqr2)(x^(1/4+sqr2)(x-8i)(i+i)

but there's no end to doing that mindless factorization

using the basic formula a^2-b^2 = (a-b)(a+b)

(don't be the 1st to be all you can, in factoring to the nth degree roots)