Question

Let P(x) = x^{​​​​​​6} + ax^{​​​​​5} + bx^{​​​​​4} + cx^​3 + dx^{​​​​​2} + ex + f be a polynomial such that P(1)=1; P(2)=2; P(3)=3; P(4)=4; P(5)=5; P(6)=6 then find the value of P(7).

Answer 1

Dear student,
In order to solve this question, by inspecting the given values of P(x)
for different values of x from 1 to 6 , we can write a function

F(x) = P(x)-x = (x-1)(x-2)(x-3)(x-4)(x-5)(x-6)
since leading coeff. has to be 1 as P(x) = x⁶ + ax^{​​​​​5} + bx^{​​​​​4} + cx^​3 + dx^{​​​​​2} + ex + f

=> p(7)-7 = 6!
=> P(7) = 727