Question

Let g(x)=x6+ax5+bx4+cx3+dx2+ex+f be a polynomial such that g(1)=1,g(2)=2,g(3)=3,g(4)=4,g(5)=5 and g(6)=6 , then the value of g(7) is​

Answer 1

Answer:

Since

f(x)−x is a polynomial of degree 6 and

has 6 roots1,2,3,4,5,6 by condition,

we can factorize f(x)−x as:

f(x)−x=C(x−1)(x−2)(x−3)(x−4)(x−5)(x−6).

Plug in x=0 in the above expression,

we have 3−0=C×6!, hence C=36!.

Therefore,

f(7)=7+(f(7)−7)=7+36!(7−1)(7−2)(7−3)(7−4)(7−5)(7−6)=7+36!×6!

=10.