Question

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

Answer 1

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