If x2 - 1 is a factor of ax4 + bx3 + cx2 + dx + e , show that a + c + e = b + d = 0. -Maths 9th

If x2 - 1 is a factor of ax4 + bx3 + cx2 + dx + e , show that a + c + e = b + d = 0. -Maths 9th
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Since x2 - 1 = (x - 1) is a factor of p(x) = ax4 + bx3  + cx2 + dx + e  ∴ p(x) is divisible by (x+1) and (x-1) separately  ⇒ p(1) = 0 and p(-1) = 0 p(1) = a(1)4 + b(1)3 + c(1)2 + d(1) + e = 0 ⇒  a + b + c + d + e = 0 ---- (i)  Similarly, p(-1) = a (-1)4 + b (-1)3  + c (-1)2 + d (-1) + e = 0 ⇒ a - b + c - d + e = 0  ⇒ a + c + e = b + d ---- (ii) Putting the value of a + c + e in eqn , we get  a + b + c + d + e = 0 ⇒ a + c + e + b + d = 0 ⇒ b + d + b + d = 0  ⇒ 2(b+d) = 0 ⇒ b + d = 0 ---- (iii) comparing equations (ii) and (iii) , we get  a + c + e = b + d = 0   
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