Question

in each case, find all polynomials of degree p <= 2 that satisfy the given conditions: p(x)=p(1-x)

Answer 1

Let p(x) = ax^2 + bx + c

Then p(1-x) = a(1-x)^2 + b(1-x) + c

Since p(x) = p(1-x), then

ax^2 + bx + c = a(1-x)^2 + b(1-x) + c

ax^2 + bx = a(1 - 2x + x^2) + b(1-x)

ax^2 + bx = a - 2ax + ax^2 + b - bx

2bx = (a + b) - 2ax

0 = (a+b) - 2(a+b)x

i.e. a+b = 0

or, a = -b

The polynomial then is p(x) = ax^2 - ax + c

So, all polynomials of the form p(x) = ax^2 - ax + c have the property that p(x) = p(1-x)