Question

Determine the polynomial equation of smallest degree whose roots are 3, 2+2i and 2-2i

Answer 1

Answer:

$P(x)=x^3-7x^2+20x-24$

Step-by-step explanation:

You are given three numbers: 3, 2+2i and 2-2i. Number 3 is a real number and numbers 2+2i and 2-2i are two conjugate complex numbers.

If the polynomial has a complex number as its root, then it also has the conjugate complex number as its root.

In your case, complex numbers 2+2i and 2-2i are conjugate to each other, so the polynomial of the smallest degree is

$P(x)=(x-3)(x-(2+2i))(x-(2-2i))=(x-3)(x^2 -(2-2i)x-(2+2i)x+(2-2i)(2+2i))=(x-3)(x^2-4x+8)=x^3-7x^2+20x-24$