Question

if two zeroes of the polynomial x³-5x²-16x+80are equal in magnitude but opposite in sign, then find the zeros​

Answer 1

Step-by-step explanation:

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