Question

Write a polynomial that will have these solutions

x=3
2+3i

Answer 1

p(x) = x³ - 7x² + 25x - 39

Explanation:

complex roots occur as conjugate pairs, thus

2 + 3i is a root then 2 - 3i is also a root

given x = 3, x = 2 +3i, x = 2 - 3i are roots, then

(x - 3), (x - (2 + 3i))(x - (2 - 3i)) are the factors and

p(x) = a(x - 3)(x - 2 - 3i)(x - 2 + 3i) ← a is a multiplier

let a = 1

p(x) = (x - 3)((x - 2)² - 9i²)

= (x - 3)(x² - 4x + 4 + 9) → [ i² = - 1 ]

= (x - 3)(x² - 4x + 13)

= x³ - 4x² + 13x - 3x² + 12x - 39

= x³ - 7x² + 25x - 39