Prove: If a is any integer and the polynomial f(x) = x 2 + ax + 1 factors (poly mod 9), then there are THREE distinct non-negative integers y less than 9 such that f(y) ≡ 0 (mod 9). Hint: If x 2 +ax+ 1 ≡ (x+b)·(x+c) (mod 9), what can you deduce about b and c? Remark: This problem is very similar to the next problem.

Dylan and dusty plan to take weekly surfing lessons together. if the 2-hour lessons are $20 per person and they plan to spend $100 each on new surfboards, what is the maximum number of lessons the two can take if the total amount spent for lessons and surfboards is at most $480?

The boiling point of water at an elevation of 0 feet is 212 degrees fahrenheit (°f). for every 1,000 feet of increase in elevation, the boiling point of water decreases by about 2°f. which of the following represents this relationship if b is the boiling point of water at an elevation of e thousand feet? a) e = 2b - 212 b) b = 2e - 212 c) e = -2b + 212 d) b = -2e + 212

Find the distance between the pair of points a(-1,8) and b(-8,4)