Question

If (x² – 1) is a factor of ax⁴ + bx³ + cx² + dx + e, then :

(a) a + c + e = 0 

(b) ace = 1 

(c) b + d = 0 

(d) Both (a) and (c)

Answer 1

(d) Both (a) and (c)

Let f(x) = ax⁴ + bx⁴ + cx² + dx + e be the given polynomial. 

Then, (x² – 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. 

Adding and subtracting the two equations, we get 

2(a + c + e) = 0 and 2(b + d) = 0 

⇒ a + c + e = 0 and b + d = 0.