The complete factorization of the polynomial, x³ + 2·x² + x + 2, obtained by finding the zeros of the polynomial is; (x + 2)·(x + i)·(x - i), which corresponds to option D
D. (x + 2)·(x + i)·(x - i)
What is a polynomial?; A polynomial consists of coefficients of variables, raised to positive integer powers, and mathematical operators, including addition subtraction and multiplication
The cubic function x³ + 2·x² + x + 2 can be factorized as follows;
When x = -2, we get;
(-2)³+ 2×(-2)² + (-2) + 2 = 0, therefore; (x + 2) is a factor of the cubic expression, x³ + 2·x² + x + 2
The division of x³ + 2·x² + x + 2 by (x + 2), indicates that we get;
x² + 1
(x + 2)| x³ + 2·x² + x + 2
x³ + 2·x²
x + 2
x + 2
0
Therefore, the factors of 2·x² + x + 2 are (x + 2) and (x² + 1)
Where x² + 1 = 0, we get;
x = √(-1)
The value √(-1) is √(-1) = i which is a complex number
x = i, therefore (x - i) is a factor of x² + 1, and we get;
(x² + 1) = (x + i) × (x - i)
2·x² + x + 2 = (x + 2) × (x² + 1)
2·x² + x + 2 = (x + 2) × (x + i) × (x - i)
The option D is correct