A polynomial P has non-negative integer coefficients. If P(1) = 8 and P(10) = 2312, what is P(2)?

9514 1404 393

  32

Step-by-step explanation:

If coefficients are positive integers, then the polynomial cannot be higher degree than 3. 10^4 = 10,000, so even a coefficient of 1 will give too large a value. P(1) = 8 tells us that the sum of coefficients must be 8, so we cannot have a degree lower than 3.

The sum of digits in 2312 is 8, so it works out that the polynomial is ...

  P(x) = 2x^3 +3x^2 +x +2 . . . . matches P(1) = 8 and P(10) = 2312

Then P(2) is ...

  P(2) = ((2·2 +3)2 +1)2 +2

  P(2) = 32

shift 4 units up, and 1 unit left.

it will translate f(x) = 3^x to g(x) = 3^(x+1) + 4