Final answer:
To write a polynomial function with given zeros, one must create factors from each zero and multiply them. The zeros 1, 1, and 2 translate into the factors (x - 1), (x - 1), and (x - 2). The correct polynomial function in standard form is Option A: (x - 1)(x - 1)(x - 2).
Step-by-step explanation:
Writing a Polynomial Function with Given Zeros
To write a polynomial function with given zeros, we use the fact that if a number is a zero of a polynomial, the factor corresponding to that zero is (x - zero). The given zeros are 1, 1, and 2; therefore, we have (x - 1), (x - 1), and (x - 2) as the factors of the polynomial.
The question provides multiple choices and asks us to select the polynomial in standard form. Multiplying the factors will give us the polynomial:
- (x - 1) times itself since the zero of 1 is repeated.
- (x - 2) for the zero of 2.
Now we multiply these factors:
(x - 1)(x - 1) = x² - 2x + 1
Then, we multiply this result by (x - 2) to get the final polynomial:
(x² - 2x + 1)(x - 2) = x³ - 2x² + x - 2x² + 4x - 2
Combine like terms to get:
x³ - 4x² + 5x - 2
This is the polynomial in standard form. Looking at the choices provided, the correct answer is (x - 1)(x - 1)(x - 2), which corresponds to Option A.