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

1.	If (x2 – 1) is a factor of ax4 + bx3 + cx2 + dx + e, show that a + c + e = b + d = 0.
Answer» Let f(x) = ax⁴ + bx³ + cx² + dx + e As (x – 1) is a factor of f(x) we have x² – 1 = (x + 1) (x – 1) hence f(1) = 0 and f(-1) = 0 f(1) = a + b + c + d + e = 0 ……………. (1) and f(-1) = a- b + c- d + e = 0 ⇒ a + c + e = b + d Substitute this value in equation (1) a + c + e + b + d=0 b + d + b + d=0 2 (b + d) = 0 ⇒ b + d = 0 ∴ a + c + e = b + d = 0

If (x2 – 1) is a factor of ax4 + bx3 + cx2 + dx + e, show that a + c + e = b + d = 0.

Answer»

Let f(x) = ax⁴ + bx³ + cx² + dx + e

As (x – 1) is a factor of f(x) we have

x² – 1 = (x + 1) (x – 1) hence f(1) = 0 and f(-1) = 0

f(1) = a + b + c + d + e = 0 ……………. (1)

and f(-1) = a- b + c- d + e = 0

⇒ a + c + e = b + d

Substitute this value in equation (1)

a + c + e + b + d=0

b + d + b + d=0

2 (b + d) = 0

⇒ b + d = 0

∴ a + c + e = b + d = 0