Question

Find a third-degree polynomial equation with rational coefficients that has roots -4 and 6 + i.

Answer 1

Answer:

x³ - 8x² - 11x + 148

Step-by-step explanation:

Given that x = 6 + i is a root then x = 6 - i is also a root

Complex roots occur as conjugate pairs.

The factors are therefore (x - (6 + i)) and(x - (6 - i))

Given x = - 4 is a root then (x + 4) is a factor

The polynomial is the product of the factors, that is

p(x) = (x + 4)(x - (6 + i))(x - (6 - i))

      = (x + 4)(x - 6 - i)(x - 6 + i)

      = (x + 4)((x - 6)² - i²)

      = (x + 4)(x² - 12x + 36 + 1)

      = (x + 4)(x² - 12x + 37) ← distribute

      = x³ + 4x² - 12x² - 48x + 37x + 148

      = x³ - 8x² - 11x + 148