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

1.	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» (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.

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»

(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.