Question

Find the remainder when 2x^10 - 3ix^8 + (1 + i)x^2 - (3 + 2i)x + 1 is divided by (x - i)

Be sure to show your work.

Answer 1

Use the polynomial remainder theorem. The remainder upon dividing a polynomial
`p(x)` by a linear factor
`x-c` is exactly
`p(c)`.

In this case,

`p(x) = 2x^(10) - 3ix^8 + (1+i)x^2 - (3+2i)x + 1`

and
`c=i`. Then the remainder is

`p(i) = 2i^(10) - 3i^9 + (1+i)i^2 - (3+2i)i + 1`

Since
`i^2=-1` and
`i^4=1`,

`p(i) = -2 - 3i - 1 - i - 3i + 2 + 1 = \boxed{-7i}`