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.

Question

dls · Answer

I am guessing we have a repeated root here..

anonymous · Answer

Yep

dls · Answer

umm so what next

anonymous · Answer

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...

dls · Answer

can we do this by graph?

dls · Answer

|dw:1379101675466:dw|
this is what the solution says..