Question

Find a cubic polynomial in standard form with real coefficients, having the given zeros -3 and 6 + 2i

Answer 1

p(x) = x³ - 9x² + 4x + 120

Explanation:

complex zeros occur in conjugate pairs

6 + 2i is a zero, hence 6 - 2i is a zero

the factors are therefore (x + 3) , (x - (6 + 2i)), (x - (6 - 2i)), hence

p(x) = (x + 3)(x - 6 - 2i)(x - 6 + 2i)

= (x + 3)((x - 6)² - 4i²)

= (x + 3)(x² - 12x + 36 + 4) ← i² = - 1

= (x + 3)(x² - 12x + 40)

= x³ - 9x² + 4x + 120 ← in standard form