How would you write the polynomial equation that has rational coefficients and the zeroes: 1+i, 4i

Answer 1

There are some conjugate pairs to keep in mind when finding complex roots.

x = 1 + i → x - (1 + i) = 0

x = 1 - i → x - (1 - i) = 0

x = 4i → x - 4i = 0

x = -4i → x + 4i = 0

(x - 4i)(x + 4i)[x - (1 - i)][x - (1 + i)] = ???

The polynomial from above is written in factored form. Let's convert it into standard form. It is easier to use the box method or some type of distributive property. Combine like terms at the end to get standard form.

(x - 4i)(x + 4i) = x² + 4ix - 4ix - 16i² = x² - 16(-1) = x² + 16 [Note: i² = -1 or i = √-1]

[x - (1 - i)][x - (1 + i)] = x² - x(1 + i) - x(1 - i) + (1 - i)(1 + i)

= x² - x - ix - x + ix + 1 - i²

= x² - 2x + 1 - (-1)

= x² - 2x + 2

x² + 16 + x² - 2x + 2 = 2x² - 2x + 18

Answer 2

P(x) = (x - (1 + i))(x - (1 - i))(x - 4i)(x + 4i)