+0

+1
54
2

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 \(g(0)\).

#1
+3

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   Dec 6, 2020
edited by CPhill  Dec 6, 2020
#2
+1

Thank you so very much for taking the time out of your day to solve my problem and even making a graph!