Question

If f(x) is a polynomial of degree 6 and f(x)=f(2-x) & f(x)=0 has 5 distinct real roots then find sum of all roots of f(x)=0.

Answer 1

I am guessing we have a repeated root here..

Answer 2

Yep

Answer 3

umm so what next

Answer 4

I'm thinking you could write the general form of \(f\) as follows:
\[f(x)=a(x-n_1)(x-n_2)(x-n_3)(x-n_4)(x-n_5)^2\]
And \(f(2-x)\) would be
\[f(2-x)=a(2-x-n_1)(2-x-n_2)(2-x-n_3)(2-x-n_4)(2-x-n_5)^2\]
Then I would think expanding the two and seeing if there's a solution to the system that you get...

Answer 5

can we do this by graph?

Answer 6

this is what the solution says..