Question

If (x^2-1) is a factor ax^4+bx^3+cx^2+dx+e then find the relation between a,b,c,d,e.

Answer 1

ANSWER :-

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

Hope it makes you clear.