Find a cubic polynomial whose zeroes are 3, 1/2, and –1.

Given:

α = 3

β = 1/2

γ = –1

Concept Used:

A cubic polynomial having α, β, and γ as zeroes is given by,

P(x) = x³ – (α + β + γ)x² + (αβ + βγ + γα)x – (αβγ)

Explanation:

α + β + γ = 3 + 1/2 –1 = 5/2

αβ + βγ + γα = 3 × 1/2 + (–1) × 1/2 + (–1) × 3

αβ + βγ + γα = 3/2 – 1/2 – 3 = –2

αβγ = 3 × 1/2 × –3 = –9/2

Putting the values, we get,

P(x) = x³ – (5/2)x² + (–2)x – (–9/2)

P(x) = x³ – 5/2x² – 2x + 9/2

Hence, P(x) = 2x³ – 5x² – 4x + 9.