Question

degree 5; zeros: 1, i, -7i

Answer 1

Since the zeros are 1, i, and -7i, the factors of the polynomial are (x - 1), (x - i), and (x + 7i).

Multiplying these factors together we get:
(x - 1)(x - i)(x + 7i)

Expanding this expression, we get:
(x^2 - x - ix + i)(x + 7i)
(x^2 - x - ix + i)(x) + (x^2 - x - ix + i)(7i)
x^3 - x^2 - ix^2 + ix + 7ix^2 - 7ix - 7i - 7i^2
x^3 - x^2 - ix^2 + ix + 7ix^2 - 7ix + 7 - 7i^2

Since i^2 = -1, we have:
x^3 - x^2 - ix^2 + ix + 7ix^2 - 7ix + 7 + 7
x^3 - x^2 - ix^2 + ix + 7ix^2 - 7ix + 14

Therefore, the polynomial with degree 5 and zeros 1, i, and -7i is:
x^3 - x^2 - ix^2 + ix + 7ix^2 - 7ix + 14