Question

Find a quadratic polynomial whose zeros are negative of the zeros of the polynomial px² + qx + r.

Answer 1

Given: Let the quadratic be px² + qx + r with p ≠ 0. Its zeros are α and β.

Step-wise calculation:

1. For px² + qx + r = 0, sum and product of zeros:

`α + β = -q/p, αβ = r/p`

2. The required zeros are –α and –β.

Their sum and product are:

`(-α) + (-β) = -(α + β) = q/p`

`(-α)(-β) = αβ = r/p`

3. A monic quadratic with these zeros is `x^2 - (q/p)x + r/p`.

Multiplying by nonzero constant k gives the general form `k(x^2 - (q/p)x + r/p)`.

Choosing k = p to clear denominators yields px² – qx + r.

A quadratic polynomial whose zeros are the negatives of the zeros of px² + qx + r is px² – qx + r or more generally `k(x^2 − (q/p)x + r/p)` for any nonzero k.