Question

if one of the zeros of cubic polynomial x^3 +ax^2+bx+c is minus 1 then the multiple of other
two zeros

Answer 1

Let α,β,ɣ are Zeros of the plynomial x3+ax2+bx+c

Given that α = -1 , βɣ = 0

α+β+ɣ = - a ⇒ β+ɣ = -a +1 ------------(1)

αβ + βɣ + αɣ = b ⇒ - β + 0 - ɣ = b ⇒ β+ɣ = -b --------------(2)

αβɣ = c ⇒ c = 0.

From (1) and (2) we get
a - b = 1

β - ɣ = √( (β+ɣ)2 - 4βɣ ) = √ b2 = b

β - ɣ = b ----------(3)

From (2) and (3) we get β = 0

then ɣ = 0 ( ∴βɣ = 0 )

from (3) ⇒ b = 0

But a - b = 1 then a =1

∴ a = 1 b = 0 c = 0