2i, 7, -7 (option C)
Explanation:
the function: f(x) = x⁴ - 45x² - 196
one of the zeros given = -2i
Note that conjugate come in pairs. So, if -2i is a zero of the function, then 2i will also be a zero of the function
The factors become:
x = -2i or x = 2i
x - (-2i) = 0 or x - 2i = 0
(x+2i)(x-2i)
we expand:
x(x-2i) + 2i(x-2i)
= x² -2xi + 2xi -4i²
= x² - 4i²
recall -1 = i²
= x² -4(-1)
= x² + 4
we need to divide x⁴ - 45x² - 196 by x² + 4 to get the other factors
We need to break down the quotient = x² - 49
The above is a difference of two squares: a² - b² = (a-b)(a+b)
x² - 49 = x² - 7²
x² - 7² = (x-7)(x+7)
x-7 = 0 or x+7 = 0
x = 7 or x = -7
All the factors of x⁴ - 45x² - 196 are -2i, 2i, +7, -7
The correct option: 2i, 7, -7 (option C)
