Let's take a closer look at the polynomial given in the problem [tex]\( x^3 + 8 \)[/tex].
### Step-by-Step Solution:
1. Understanding the Polynomials:
- We need to factor [tex]\( x^3 + 8 \)[/tex] into a product of polynomials.
2. Recognize the Sum of Cubes Formula:
- The expression [tex]\( x^3 + 8 \)[/tex] fits the form of a well-known algebraic identity for the sum of cubes: [tex]\( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \)[/tex].
- Here, [tex]\( a = x \)[/tex] and [tex]\( b = 2 \)[/tex], giving us:
[tex]\[ x^3 + 8 = x^3 + 2^3 \][/tex]
3. Apply the Sum of Cubes Formula:
- Using the sum of cubes formula, we have:
[tex]\[
x^3 + 2^3 = (x + 2)\left(x^2 - x \cdot 2 + 2^2\right)
\][/tex]
- Simplifying this, we get:
[tex]\[
x^3 + 8 = (x + 2)(x^2 - 2x + 4)
\][/tex]
4. Check the Factored Form:
- Now we can verify the provided solutions:
- [tex]\((x + 2)(x^2 - 2x + 4)\)[/tex]
- [tex]\((x - 2)(x^2 + 2x + 4)\)[/tex]
- [tex]\((x + 2)(x^2 - 2x + 8)\)[/tex]
- [tex]\((x - 2)(x^2 + 2x + 8)\)[/tex]
- From our earlier calculations, we know that the correct factored form matches:
[tex]\[
(x + 2)(x^2 - 2x + 4)
\][/tex]
5. Conclusion:
- Therefore, the polynomial [tex]\( x^3 + 8 \)[/tex] factors correctly to:
[tex]\[
(x + 2)(x^2 - 2x + 4)
\][/tex]
### Final Answer:
The polynomial [tex]\( x^3 + 8 \)[/tex] is equal to [tex]\((x + 2)(x^2 - 2x + 4)\)[/tex].
