Q. 55.0( 2 Votes )
Prove that if x2 – 1 is a factor of ax3 + bx3 + cx + d, then a = –c and b = –d.
Answer :
Given, ax3 + bx2 + cx + d
Need to show a = – c and b = – d if x2 – 1 is a factor
⇒ consider, x2 – 1 is a factor of ax3 + bx2 + cx + d
⇒ then x = + 1, x = – 1
⇒ substitute x value in the equation ax3 + bx2 + cx + d
We get as follows
⇒ for x = 1 we get a(1)3 + b(1)2 + c(1) + d
= a + b + c + d ……..eq(1)
⇒ for x = – 1 we get a(– 1)3 + b(– 1)2 + c(– 1) + d
= – a + b – c + d ………eq(2)
⇒ Solving the two equations we get
⇒ (a + b + c + d) + (– a + b – c + d) = 0
⇒ b + d = 0
∴ b = – d
And if (a + b + c + d) – (– a + b – c + d) = 0
⇒ a + c = 0
∴ a = – c
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