pure maths polynomial 短題目prove

If part:

Let α, β and γ be the roots, then consider when we sub x = -p into the equation:

(-p)3 + p(-p)2 - qp + r = -p3 + p3 - qp + pq (since r = pq)

= 0

Therefore -p is a root of the equation.

Also, by comparing coefficient in the relation:

x3 + px2 + qx + r = (x - α)(x - β)(x - γ)

We have -p = α + β + γ, hence without loss of generality, we can assume that -p = α

Then we have β + γ = 0, i.e. sum of two roots is zero

Only if part:

Let α, β and -α be the roots, then by comparing coefficient in the relation:

x3 + px2 + qx + r = (x - α)(x - β)(x + α)

We have:

-p = β

-α2 = q

α2β = r

Hence pq = α2β = r