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To write a polynomial function with the given zeros, we need to consider the fact that complex zeros occur in conjugate pairs. Since 2 and 3 are real zeros, we have the factors (x - 2) and (x - 3) in the polynomial.
The imaginary zero i occurs with its conjugate -i. Therefore, we have the factors (x - i) and (x + i) in the polynomial.
To find the polynomial function, we multiply all these factors together:
(x - 2)(x - 3)(x - i)(x + i)
Expanding this multiplication will give us the desired polynomial.
(x - 2)(x - 3)(x - i)(x + i)
= (x^2 - 5x + 6)(x^2 + 1)
= (x^4 - 4x^3 - x^2 + 6x^2 - 5x + 6)
= x^4 - 4x^3 + 5x^2 - 5x + 6
Hence, the polynomial function of minimum degree in standard form with real coefficients and the given zeros is:
f(x) = x^4 - 4x^3 + 5x^2 - 5x + 6