Achilles ordered a pizza with 16 slices. Every hour, he ate half of the remaining slices. Let f(n) be the number of slices Achilles ate in the hour since he got home.

1. when "a" is a root of a polynomial, (x -a) is a factor of it. for the three roots given, the factors of the desired polynomial are (x +2)(x -1/2)(x -3).

in order to make the leading coefficient be negative, we need to multiply this product by a negative number. any negative number will do, but we choose a small (magnitude) value that will eliminate the fraction: -2.

then

f(x) = -2(x +2)(x -1/2)(x -3) = -(x +2)(2x -1)(x -3)

= -(2x² +3x -2)(x -3)

= -(2x³ -3x² -11x +6)

f(x) = -2x³ +3x² +11x -6

2. a graph created by the desmos on-line graphing calculator is shown, and the zeros are highlighted.

3. as indicated in part 1, the multiplier of this equation can be anything and the zeros will remain the same. you want a negative leading coefficient, so the "anything" is restricted to any of the infinite number of numbers that will make that be the case.

step-by-step explanation:

the answer to this question depends on certain assumptions.   if the 6-pint jar starts out empty, and the 3-pint jar full, pouring the 3 pints into the 6-pint jar results in a half-full 6-pint jar.   if we refill the 3-pint jar and then pour its contents into the half-full 6-pint jar, then the 6-pint jar will be full.