Suppose is a polynomial of degree five for which \(g(1) = 2, g(2) = 3, g(3) = 4, g(4) = 5, g(5) = 6\) and \(g(6) = -113. \) Find \(g(0)\).
Very difficult (if not impossible) to detect any patterns for this, phmqt !!!!
We have this cumbersome system
a + b + c + d + e + f = 2
32a + 16b + 8c + 4d + 2e + f = 3
243a + 81b + 27c + 9d +3e + f = 4
1024a + 256b + 64c + 16d + 4e + f =5
3125 a + 625b + 125c + 25d + 5e + f = 6
7776a + 1296b + 216c + 36d + 6e + f = -113
This is very time-consuming to solve (but it's do-able)...so...I will use some tecnology to find f (what we are really looking for )
The polynomial is
a b c d e f
-1x^5 + 15x^4 - 85x^3 +225x^2 -273x + 121
g(0) = f = 121
Verified with Desmos, here : https://www.desmos.com/calculator/4dm3cbtuxo