Final answer:
To find the remaining zeros of the polynomial function f(x) = x³ - 2x² + 9x - 18 with a given zero of 3i, we first recognize that -3i is also a zero due to the Complex Conjugate Root Theorem. We then divide f(x) by x² + 9 and solve the resulting linear equation to find the last zero.
Step-by-step explanation:
The question asks to use the given zero of a polynomial function to find the remaining zeros.
The polynomial is given as f(x) = x³ - 2x² + 9x - 18, and one zero is provided, which is 3i.
According to the Complex Conjugate Root Theorem, if 3i is a zero, then its complex conjugate, -3i, is also a zero of f(x).
To find the remaining zeros, we can use polynomial division or synthetic division to divide the polynomial by the quadratic factor corresponding to these two complex roots, (x - 3i)(x + 3i) = x² + 9, to determine the remaining real zero.
Dividing f(x) by x² + 9, we get a linear factor. We can then set this factor equal to zero to solve the mathematical problem completely and find the last remaining zero of the polynomial function.