Question

What is the complete factorization of the polynomial below?

Answer 1

Option A: $(x+1)(x+3 i)(x-3 i)$ is the complete factorization of the polynomial $x^{3} +x^{2} +9x+9$

Explanation:

The polynomial is $x^{3} +x^{2} +9x+9$

Now, we shall find the complete factorization of the polynomial.

Let us group the common terms, we have,

$\left(x^{3}+x^{2}\right)+(9 x+9)$

Taking out the common terms,

$x^{2} (x+1)+9(x+1)$

Factor out $x+1$ from both the terms, we have,

$\left(x^{2}+9\right)(x+1)=0$

The term $\left(x^{2}+9)\right.$ can be factored as

$\begin{aligned}x^{2}+9 &=0 \\x^{2} &=-9 \\x &=\pm 3 i\end{aligned}$  and $\begin{aligned}x+1 &=0 \\x &=-1\end{aligned}$

Thus, the roots are $x=\pm 3 i$ and $x=-1$

These roots can be written as $(x+1)(x+3 i)(x-3 i)$

Thus, the complete factorization of the polynomial is $(x+1)(x+3 i)(x-3 i)$

Answer 2

Answer: A

Step-by-step explanation:

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