can u answer this question and give me step by step solution
Factor the expression x 3 + 8 as completely as possible.

To factor the expression \( x^3 + 8 \), we can recognize this as a sum of cubes.

The formula for factoring a sum of cubes is:
\[
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
\]
In this case:
- \( a = x \)
- \( b = 2 \) (since \( 2^3 = 8 \))

Now, let's follow the steps to factor the expression \( x^3 + 8 \):

### Step 1: Apply the sum of cubes formula
Using the formula \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \):

\[
x^3 + 8 = (x + 2)((x)^2 - (x)(2) + (2)^2)
\]

### Step 2: Simplify the second factor
Now, simplify the expression inside the second parentheses:

\[
(x)^2 = x^2
\]
\[
-(x)(2) = -2x
\]
\[
(2)^2 = 4
\]

So, the factored form is:

\[
x^3 + 8 = (x + 2)(x^2 - 2x + 4)
\]

### Final answer:
The completely factored form of \( x^3 + 8 \) is:
\[
(x + 2)(x^2 - 2x + 4)
\]

Would you like more details or have any questions?

Here are five related questions to practice:

1. How do you factor \( x^3 - 27 \)?
2. Factor \( 8x^3 + 27 \).
3. What is the sum of cubes formula, and how does it differ from the difference of cubes formula?
4. Factor the expression \( x^3 + 27 \).
5. Solve the equation \( x^3 + 8 = 0 \).

**Tip:** Always check for special factoring patterns, such as the sum or difference of cubes, to simplify problems quickly.

Factor the expression x^3 + 8 as completely as possible.

Learn how to factor the expression x^3 + 8 using the sum of cubes formula. This detailed step-by-step solution covers algebraic concepts suitable for students in Grades 9-12, including factoring techniques for polynomials.

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