Final answer:
To create a polynomial function in standard form with given zeros, convert each zero into a factor and multiply them together. The zeros include a complex number and its conjugate, resulting in a polynomial that is expanded and simplified to reach standard form.
Step-by-step explanation:
To write a polynomial function in standard form given a set of zeros, we start by turning each zero into a factor of the function. Given the zeros x = 2-√ and x = -i, we first address the complex zero. Since complex roots come in conjugate pairs, if -i is a zero, then its conjugate, +i, is also a root. Thus, we have three roots total: x = 2-√, x = -i, and x = +i.
The factors corresponding to these roots are (x - (2-√)), (x + i), and (x - i).
Now, we can multiply these factors together to find the polynomial:
- (x - (2-√)) * (x + i) * (x - i)
Using the fact that i2 = -1 and multiplying the complex factors first:
- (x + i) * (x - i) = x2 + 1
Next, we multiply this result by (x - (2-√)):
This multiplication results in a polynomial, which is in standard form once fully expanded and simplified.