Final answer:
The polynomial function for the set of zeros x=2, i, and −i is found by multiplying the factors (x-2), (x-i), and (x+i), resulting in the standard form of the polynomial: x³ - 2x² + x - 2.
Step-by-step explanation:
To write a polynomial function in standard form for the set of zeros given (x=2, i, −i), we start by using the fact that if x=c is a zero of the function, then (x-c) is a factor of that function.
Since the zeros include complex numbers, and the coefficients of our polynomial are real numbers, the polynomial must have the complex conjugate pair i and −i as factors.
For x=2, the factor is (x-2). For the zeros i and −i, the factors are (x-i) and (x+i) respectively. We then multiply these factors to find the polynomial in standard form:
(x-2)(x-i)(x+i) = (x-2)((x-i)(x+i))
= (x-2)(x²+1)
Now we multiply the remaining factors:
(x-2)x² + (x-2)1 = x³ - 2x² + x - 2
Therefore, the polynomial function in standard form is x³ - 2x² + x - 2.