Let's factor the polynomial [tex]\( x^3 + 3x^2 + 9x + 27 \)[/tex] step-by-step.
Firstly, we look at the polynomial:
[tex]\[ x^3 + 3x^2 + 9x + 27 \][/tex]
We want to see if we can simplify this polynomial into a product of lower-degree polynomials. Factoring polynomials typically involves finding patterns or using methods such as synthetic division or recognizing common polynomial identities.
In this case, we start by grouping the terms in a way that might reveal a common factor or pattern:
[tex]\[ x^3 + 3x^2 + 9x + 27 = (x^3 + 3x^2) + (9x + 27) \][/tex]
Next, let's factor out the greatest common factors from both groups:
[tex]\[ x^3 + 3x^2 + 9x + 27 = x^2(x + 3) + 9(x + 3) \][/tex]
Notice that [tex]\( (x + 3) \)[/tex] is a common factor in both terms:
[tex]\[ x^3 + 3x^2 + 9x + 27 = (x^2 + 9)(x + 3) \][/tex]
Now, examine [tex]\( x^2 + 9 \)[/tex] to see if it can be factored further. Since [tex]\( x^2 + 9 = x^2 + (3i)^2 \)[/tex] for which completing the square doesn't introduce any real roots, we note that it is irreducible over the real numbers.
Therefore, the complete factorization of [tex]\( x^3 + 3x^2 + 9x + 27 \)[/tex] over the complex numbers is:
[tex]\[ x^3 + 3x^2 + 9x + 27 = (x + 3)(x^2 + 9) \][/tex]
Simplifying the answer, the quadratic term [tex]\( x^2 + 9 \)[/tex] can be viewed as having imaginary roots. However, our focus for this solution is the primary factorization:
[tex]\[ (x + 3) (x^2 + 9) \][/tex]
Looking at the answer choices:
A. [tex]\((x-3)(x+3i)(x-3i)\)[/tex] does not match our factorized terms.
B. [tex]\((x-3)(x+3)(x+3)\)[/tex] does not match.
C. [tex]\((x+3)(x+3i)(x-3i)\)[/tex] aligns with our factors correctly.
D. [tex]\((x+3)(x+3)(x+31)\)[/tex] does not match.
Thus, the correct answer is:
[tex]\[ \boxed{(x+3)(x+3i)(x-3i)} \][/tex]
