The restaurant sold coffee and tea during the breakfast shift. The number of cups of tea they sold was 6 more than 1/3 of the number of cups of coffee sold. If the total cups of coffee and tea sold was 70, how many cups of each were sold?

cups of tea = 22

cups of coffee = 48

Step-by-step explanation:

c = coffee

t (tea) = 6 + 1/3c

t + c = 70

c + 6 + 1/3c = 70

4/3c + 6 = 70

4/3c = 64

4c = 192

c = 48

t = 70 - 48 = 22

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.