To factor the polynomial [tex]\(x^4 + 4\)[/tex], we will follow a structured algebraic approach.
Firstly, let's rewrite the polynomial in a form that can help us recognize a pattern or make the factorization easier.
[tex]\[ x^4 + 4 \][/tex]
We will attempt to factor it by recognizing how it corresponds to a sum of squares or other special forms.
1. Recognize Special Polynomial Forms:
Notice that [tex]\(x^4 + 4\)[/tex] can be analyzed similarly to how we deal with expressions involving squares, although it is not a simple sum of squares. We can approach this by paying attention to pairs of terms that can possibly be rearranged or combined into a known factorable form.
2. Intermediate Factorization (Using Substitution):
To see better, add and subtract a term that complements to form perfect squares or differences where necessary. Here, we can use the following clever trick:
[tex]\[ x^4 + 4 = x^4 + 4 \cdot 1^2 \][/tex]
Then, recognize the structure related to special binomials. Recall the factorization for terms involving squared terms which ultimately come close to forms of complex roots.
3. Factor by Grouping:
Notice the expression is factorable by a method related to grouping:
[tex]\[ x^4 + 4 = (x^2)^2 + (2)^2 \][/tex]
4. Applying the Sum of Squares and Related Factoring Methods:
We need to think of both terms squared plus twice their product method commonly used but directly we go into recognizing complex conjugates and group for quadratic forms:
[tex]\[ x^4 + 4 \][/tex]
Recognize that:
[tex]\[ x^4 + 4 = (x^2 - 2x + 2)(x^2 + 2x + 2) \][/tex]
5. Verification:
Let's verify by expanding [tex]\((x^2 - 2x + 2)(x^2 + 2x + 2)\)[/tex]:
[tex]\[
\begin{aligned}
(x^2 - 2x + 2)(x^2 + 2x + 2) &= (x^2 \cdot x^2 + x^2 \cdot 2x + x^2 \cdot 2) + ((-2x) \cdot x^2 + (-2x) \cdot 2x + (-2x) \cdot 2) + (2 \cdot x^2 + 2 \cdot 2x + 2 \cdot 2) \\
&= x^4 + 2x^3 + 2x^2 - 2x^3 - 4x^2 - 4x + 2x^2 + 4x + 4 \\
&= x^4 + 0x^3 + 0x^2 + 0x + 4 \\
&= x^4 + 4
\end{aligned}
\][/tex]
Thus, confirming our factorization, the final factors of the polynomial [tex]\(x^4 + 4\)[/tex] are:
[tex]\[
(x^2 - 2x + 2)(x^2 + 2x + 2)
\][/tex]
