Final answer:
The correct polynomial equation with roots -1, i, and -i is (x + 1)(x - i)(x + i), which expands to (x + 1)(x² + 1). This equation satisfies the given roots through the relationship between roots and factors of a polynomial.
Step-by-step explanation:
The polynomial equation of least degree with roots -1, i, and -i is given by option b): (x + 1)(x - i)(x + i). This is because the product of these factors will yield a polynomial equation with the aforementioned roots.
To explain this, remember that the factors of a polynomial are directly related to its roots: for a root a, the factor is (x - a). The root -1 leads to the factor (x + 1), while the imaginary roots i and -i lead to the factors (x - i) and (x + i), respectively. The polynomial equation of least degree with roots -1, i, and -i is option b) (x + 1)(x - i)(x + i). This equation represents the product of the factors (x + 1), (x - i), and (x + i), which will give the desired roots when solved.
When multiplied, (x - i) and (x + i) give us x² + 1, since the product includes the difference of squares. Combining this with the factor (x + 1) from the first root, we get the correct polynomial equation (x + 1)(x² + 1).