Option A: [tex](x+1)(x+3 i)(x-3 i)[/tex] is the complete factorization of the polynomial [tex]x^{3} +x^{2} +9x+9[/tex]
Explanation:
The polynomial is [tex]x^{3} +x^{2} +9x+9[/tex]
Now, we shall find the complete factorization of the polynomial.
Let us group the common terms, we have,
[tex]\left(x^{3}+x^{2}\right)+(9 x+9)[/tex]
Taking out the common terms,
[tex]x^{2} (x+1)+9(x+1)[/tex]
Factor out [tex]x+1[/tex] from both the terms, we have,
[tex]\left(x^{2}+9\right)(x+1)=0[/tex]
The term [tex]\left(x^{2}+9)\right.[/tex] can be factored as
[tex]\begin{aligned}x^{2}+9 &=0 \\x^{2} &=-9 \\x &=\pm 3 i\end{aligned}[/tex] and [tex]\begin{aligned}x+1 &=0 \\x &=-1\end{aligned}[/tex]
Thus, the roots are [tex]x=\pm 3 i[/tex] and [tex]x=-1[/tex]
These roots can be written as [tex](x+1)(x+3 i)(x-3 i)[/tex]
Thus, the complete factorization of the polynomial is [tex](x+1)(x+3 i)(x-3 i)[/tex]
