Question

check whether z⁵ - z² + 2z³ + 1 is exactly divisible by z + 1 or not.​

Answer 1

Step-by-step explanation:

p(x) = z⁵ - z² + 2z³ + 1

g(x) = z + 1

if p(x) is exactly divisible by g(x), the remainder has to be zero. we can use remainder theorem to find the remainder.

let g(x) = 0 => z + 1 = 0 => z = - 1

now p(- 1) has to be zero if p(x) is exactly divisible by g(x)

p(-1) = (-1)⁵ - (-1)² + 2(-1)³ + 1

=> p(-1) = -1 -1 - 2 + 1

=> p(-1) = - 4 + 1 => - 3

as p(-1) is not zero, p(x) is not exactly divisible by g(x)