Final answer:
To find the values of a for which ai is a root of the given complex polynomial, one must realize that complex roots occur in conjugate pairs.
Step-by-step explanation:
The student is asked to find the values of a ∈ R for which ai is a solution of z⁴−2z³+7z²−4z+10=0. To solve complex equations such as this, where i is the imaginary unit, we can use the fact that complex roots of polynomials with real coefficients come in conjugate pairs. Therefore, if ai is a root, its conjugate -ai must also be a root.
To find all roots of this polynomial, we first need to use synthetic division or polynomial division to simplify the equation using the suspected roots ai and -ai. After finding a, we can factor the polynomial or use numerical methods to find the remaining roots of the equation. As this polynomial is a quartic, there will be four roots in total. After calculating all the roots, it is crucial to check if the solutions are reasonable.
The information above about quadratic equations seems to be another topic, but we can draw parallels. The process involves using the quadratic formula, x = (-b ± √(b² - 4ac)) / (2a), after rearranging the equation to have 0 on one side. This method is extremely useful for solving quadratic equations and implies that to simplify algebra, we should always eliminate terms wherever possible.