Question

What is the complete factorization of the polynomial below ?

Answer 1

(x - 2i)(x + 2i)(x + 1)

Explanation:

Factor x³ + x² + 4x + 4.

Note that x² is common to the first two terms, and that 4 is common to the last two terms.

Thus: x³ + x² + 4x + 4 = x²(x + 1) + 4(x + 1).

We see that x + 1 is common to both terms. Thus, we have:

(x² + 4)(x + 1).

Note that x² + 4 has two imaginary roots: 2i and -2i. Thus, the complete

factorization of the polynomial is (x - 2i)(x + 2i)(x + 1).

Answer 2

`(x+1)(x+2i)(x-2i)`

Explanation:

`x^3+x^2+4x+4`

Factor the given polynomial

Group first two terms and last two terms

`(x^3+x^2)+(4x+4)`

Factor out GCF from each group

`x^2(x+1)+4(x+1)`

Factor out x+1

`(x^2+4)(x+1)`

Now factor out x^2+4 that is x^2 + 2^2

`x^2+4= (x+2i)(x-2i)`

`(x+1)(x+2i)(x-2i)`