Question

Find a cubic polynomial whose zeros are 3, 5 and –2.

Answer 1

Given: Zeros: 3, 5 and –2.

Step-wise calculation:

1. A cubic polynomial with zeros α, β, γ is p(x) = x³ – (α + β + γ)x² + (αβ + βγ + γα)x – (αβγ).

2. Compute the symmetric sums:

α + β + γ = 3 + 5 + (–2)

= 6

αβ + βγ + γα = (3 × 5) + (5 × –2) + (–2 × 3) 

= 15 – 10 – 6 

= –1

αβγ = 3 × 5 × (–2) 

= –30

3. Substitute into the formula:

p(x) = x³ – 6x² + (–1)x – (–30) 

= x³ – 6x² – x + 30

A cubic polynomial with zeros 3, 5 and –2 is p(x) = x³ – 6x² – x + 30.