1. Let f(x) be a polynomial such that f(0) = 4, f(1) = 5, and f(2) = 10. Find the remainder when f(x) is divided by x(x - 1)(x - 2).
2. Find a polynomial f(x) of degree 5 such that both of these properties hold:
f(x) is divisible by x^3. (so its in ax^5+bx^4+cx^3 form)
f(x) + 2 is divisible by (x+1)^3.
For 1 see https://web2.0calc.com/questions/let-f-x-be-a-polynomial-such-that-f-0-4-f-1-5-and_1#r2
Since f(x) is divisible by x^3, f(x) is of the form ax^5 + bx^4 + cx^3.
You then want ax^5 + bx^4 + cx^3 + 2 to be divisible by (x + 1)^3. Using long division, you get the equations
-10a + 6b - 3c = 0
4a - 3b + 2c = 0
-a + b - c + 2 = 0
==> a = 6, b = 16, c = 12
So f(x) = 6x^5 + 16x^4 + 12x^3.