Question

find the zeroes of the polynomial x3-5x2-16x+80 if its two zeroes are equal in magnitude but oppsite 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                

Answer 2

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