Question

Find zeros of f(x)= x³ - 5x² - 16x + 80 , if it’s two zeros are equal in magnitude but opposite in sign

Answer 1

Let α, β, Δ be the roots of given polynomial x3-5x2-16x+80.
Given 2 of its roots are equal but of opposite signs
Let α,β be equal roots
such that β = -α

From this polynomial,
Sum of the roots ⇒ α+β+Δ = -coefficient of x²/ coefficient of x³
⇒ α-α+Δ = -(-5)/1
Δ = 5 --------- One of the root (1)

Product of roots ⇒ αβΔ = -constant/coefficient of x³
⇒ α(-α)Δ = -80/1
⇒ -α²Δ = -80
⇒ α²Δ = 80

From(1), Substituting Δ=5
⇒ α²×5 = 80
⇒ α² = 80/5
⇒ α² = 16
⇒ α = √16 = +-4
⇒ α = +4 or -4 {where -4 = -α = β}

Therefore, the three roots of given polynomial x3-5x2-16x+80 α,β,Δ are 4,-4,5