Find 3rd degree function with real coefficients, if the roots are -3, 1-2i

Question

campbell_st · Answer

for the roots you'll have 

(x - 1)^2 = -4

to check 

$$x -1 = \sqrt{-4}$$

then 

$$x - 1 = \sqrt{2i^2}$$

$$x - 1 = \pm 2i$$

so 

$$x = 1 \pm 2i$$

so going back 

(x -1)^2 = 4

is the same as 

x^2 - 2x + 1 - 4

so 

(x^2 - 2x - 3) is a quadratic factor

the other factor is (x + 3)

so the factored polynomial is

$$P(x) = (x + 3)(x^2 - 2x - 3) $$

just distribute to get the cubic equation