Your younger sibling runs up to you and excitedly exclaims, “I’m thinking of a number. If I add it to the number 2 ten times, that is, 2 + my number + my number + my number, and so on, then the answer is 2 . What is my number?” You almost immediately answer, “zero,” but are you sure? Can you find a different number (other than zero) that has the same property? If not, can you justify that your answer is the only correct answer? Please help me this is confusing

Yes you are sure... No matter what other number you replace that with it will have a sum of more than 2. 2 +0^10= 2... 0 is the only number that will give you the sum of 2

Step-by-step explanation:

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.