Final answer:
The polynomial x^3 - 4x^2 + 9x - 10 divided by x - 2 simplifies to x^2 - 2x + 5 by using synthetic division, which corresponds to option (a).
Step-by-step explanation:
The main answer to the student's question of simplifying the polynomial x^3 - 4x^2 + 9x - 10 divided by x - 2 is found through long division or synthetic division. Let's use synthetic division for efficiency. We write down the coefficients of the polynomial which are 1 for x^3, -4 for x^2, 9 for x, and -10 for the constant term. Then, we write 2 to the left, representing the zero of our divisor x - 2. We bring down the 1, multiply it by 2 to get 2, and write this under the -4, obtaining -2. Continuing this process, we multiply -2 by 2 to get -4, combine it with 9 to get 5, multiply 5 by 2 to get 10, and add this to -10 to get 0, showing there's no remainder. The result is the simplified polynomial x^2 - 2x + 5, represented by the new coefficients, which is option (a).