Suppose g(x) 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).
Since g(x) is a polynomial of degree five, g(x) = ax^5 + bx^4 + cx^3 + dx^2 + ex + f. Then a + b + c + d + e + f = 2, 32a + 16b + 8c + 4d + 2e + f = 3, and so on. Solving, we get a = -1/30, b = 1/2, c = -17/6, d = 15/2, e = -122/5, and f = 5, so g(0) = 5.
+118608 \(g(x)=ax^5+bx^4+cx^3+dx^2+ex+f \\~\\ Find \;\;f\\~\\ g(1)=a+b+c+d+e+f =2\\ g(2)=32a+16b+8c+4d+2e+f =3\\ g(3)=243a+81b+27c+9d+3e+f =4\\ etc\\ \)
This appears to me to be a very long matrix problem (if done by hand)
I am wondering if there is a shortcut that I have not seen.....
CODING
g(x)=ax^5+bx^4+cx^3+dx^2+ex+f \\~\\
Find \;\;f\\~\\
g(1)=a+b+c+d+e+f =2\\
g(2)=32a+16b+8c+4d+2e+f =3\\
g(3)=243a+81b+27c+9d+3e+f =4\\
etc\\
