If `x^2-1` is `a` factor of `a x^4+b x^3+c x^2+dx+e ,` show that `a+c+e=b+d`

Explanation

Let `f(x)=ax^4+bx^3+cx^2+dx+e` be the given polynomial.
Then, `(x^2–1)` is a factor of `f(x)`
`⇒(x–1)(x+1)` is a factor of `f(x)`
`⇒(x–1)` and `(x+1)` are factors of `f(x)`
`⇒f(1) = 0` and `f(-1)=0`
`⇒a+b+c+d+e=0` and `a-b+c–d+e=0`
Comparing two equations, we get:
`=>2(a+c+e)=2(b+d)`
`=>a+c+e=b+d`
Hence proved.