To factorize the polynomial [tex]\( x^3 + x^2 + 4x + 4 \)[/tex], we need to break it down into a product of simpler polynomials or factors.
The polynomial is given as:
[tex]\[
x^3 + x^2 + 4x + 4
\][/tex]
After carefully examining and performing factorization, the polynomial can be expressed as:
[tex]\[
(x + 1)(x^2 + 4)
\][/tex]
Let's analyze this factorization step-by-step:
1. The first factor is [tex]\( (x + 1) \)[/tex].
2. The second factor [tex]\( (x^2 + 4) \)[/tex] is a quadratic expression.
To verify, we can expand the factors to check if they produce the original polynomial:
[tex]\[
(x + 1)(x^2 + 4) = x(x^2 + 4) + 1(x^2 + 4) = x^3 + 4x + x^2 + 4 = x^3 + x^2 + 4x + 4
\][/tex]
This confirms that our factorization is correct:
[tex]\[
x^3 + x^2 + 4x + 4 = (x + 1)(x^2 + 4)
\][/tex]
Now, let's compare this with the given options:
A. [tex]\((x-1)(x+2i)(\pi+2i)\)[/tex]
B. [tex]\((x+1)(x+2i)(x+2i)\)[/tex]
C. [tex]\((x-1)(x+2i)(x-2i)\)[/tex]
D. [tex]\((x+1)(x+2i)(x-2i)\)[/tex]
Among these options, we need to identify which one matches our factorization. We can see that:
[tex]\[
(x^2 + 4) = (x^2 - (-4)) = (x^2 - (2i)^2) = (x + 2i)(x - 2i)
\][/tex]
Hence,
[tex]\[
(x + 1)(x^2 + 4) = (x + 1)(x + 2i)(x - 2i)
\][/tex]
Thus, the option that correctly represents the complete factorization is:
D. [tex]\( (x+1)(x+2i)(x-2i) \)[/tex]
This matches our factorized form perfectly, showing that the complete factorization of the polynomial [tex]\( x^3 + x^2 + 4x + 4 \)[/tex] is indeed [tex]\((x + 1)(x + 2i)(x - 2i)\)[/tex].
