Question

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

Answer 1

Step-by-step explanation:

Given : x 2 – 1 is a factor of f(x) = ax 4 + bx 3 +cx 2 + dx + e

Now (x 2 – 1) can be written as (x + 1) (x – 1)

⇒ (x – 1) and (x + 1) are factors of f(x)

⇒ f(–1) = 0

⇒ a(–1)4 + b(–1)3 + c(–1)2 + d(–1) + e = 0

⇒ a – b + c – d + e = 0

⇒ a + c + e = b + d ..... (1)

and f (1) = 0

⇒ a(1)4 + b(1)3 + c(1)2 + d(1) + e = 0

⇒ a + b + c + d + e = 0

⇒ b + d + b + d = 0 ( from (1) )

⇒ 2 (b + d) = 0

⇒ b + d = 0 ..... (2)

from (1) and (2) we get

a + c + e = b + d = 0