Write a polynomial function in standard form for the set of zeros. Show your step-by-step work. x=2, i, −i
To start, we know that if x=2 is a zero of the polynomial function, then (x - 2) must be a factor of the polynomial.
Thus, one factor of the polynomial is (x - 2).
We also know that if x=i is a zero of the polynomial function, then (x - i) must be a factor of the polynomial.
Thus, another factor of the polynomial is (x - i).
Similarly, if x=-i is a zero of the polynomial function, then (x + i) must be a factor of the polynomial.
Thus, the last factor of the polynomial is (x + i).
Now, multiplying these three factors together:
(x - 2)(x - i)(x + i) = (x - 2)(x^2 - i^2)
= (x - 2)(x^2 - (-1))
= (x - 2)(x^2 + 1)
Expanding this further gives:
(x - 2)(x^2 + 1) = x(x^2 + 1) - 2(x^2 + 1)
= x^3 + x - 2x^2 - 2
= x^3 - 2x^2 + x - 2
Therefore, the polynomial function in standard form with the set of zeros x=2, i, −i is:
f(x) = x^3 - 2x^2 + x - 2.