If $$x^2-1$$​ is a factor of $$ax^4+bx^3+cx^2+dx+e$$​, then which of the following is a possible relation between the coefficients of powers of $$x$$​​

Correct option is A

Given:

$$x^2 - 1$$​ is a factor of $$ax^4 + bx^3 + cx^2 + dx + e$$​​

Solution:
Substitute x = 1 and x = -1 in the polynomial:

​$$f(1) = a(1)^4 + b(1)^3 + c(1)^2 + d(1) + e = a + b + c + d + e = 0$$​​

$$f(-1) = a(-1)^4 + b(-1)^3 + c(-1)^2 + d(-1) + e = a - b + c - d + e = 0$$​

a + b + c + d + e = 0

a - b + c - d + e = 0

Adding the two equations:

2a + 2c + 2e = $$0 \Rightarrow a + c + e$$ = 0

Subtracting the second from the first:

2b + 2d = $$0 \Rightarrow b$$​ + d = 0

a + c + e = b + d