Since k = 2i is a given zero of f(x), it means that x = 2i is a root of the polynomial. To factor the polynomial completely, we can perform polynomial division of f(x) by the corresponding factor, (x - 2i).
The polynomial f(x) is given by f(x) = x³ + (7−2i)x² + (10−14i)x − 20i.
Let's use polynomial division to divide f(x) by (x - 2i):
1. Divide the leading term of f(x) (which is x³) by the leading term of (x - 2i) (which is x). This gives us x². Write this above the division bar.
2. Multiply (x - 2i) by x² and subtract the result from f(x) to obtain the new polynomial. We get (7 - 2i)x² + (10 - 14i)x − 20i - [(x²*(x - 2i))]) = (9 + 4i)x - 20i.
3. Repeat steps 1 and 2 until the degree of the new polynomial is less than the degree of the divisor (x - 2i).
Performing these steps, the quotient we obtain is x² + (4i + 1)x + 5i, and the remainder is 0.
Since (x-2i) divides f(x) evenly (remainder is zero), we have that (x-2i) is a factor of f(x). Therefore, the factorized form of the polynomial is given by f(x) = (x - 2i) * (x² + (4i + 1)x + 5i).
This means that the numerical factorization of the polynomial f(x), given that k = 2i is a root of f(x), is (x - 2i) multiplied by the quotient (x² + (4i + 1)x + 5i). This completes the process of factoring f(x) completely.