Question

(x²-1) is a factor of ax power 4+bx³+cx²+dx+e show that a+c +e=b+d=0

Answer 1

x^2-1=0 because it is a. factor so there fore x=1 substitute in given equation you will get a+b+c+d+e=0 hence proved

Answer 2

Answer:

Let p(x) = ax⁴ + bx³ + cx² + dx + e be the given polynomial. Then,

(x² - 1) is a factor of p(x).

=> (x-1) (x+1) is a factor of p(x).

=> (x-1) and (x+1) are factors of p(x).

=> p(1) = 0 and p(-1) = 0.

=> a + b + c + d + e = 0 and a - b + c - d + e = 0.

Adding and subtracting these two equations , we get

2 (a+c+e) = 0 and 2 (b+d) = 0

=> a + c + e = 0 and b + d = 0

=> a + c + e = b + d = 0.