Il polinomio 27x3 + 8 si fattorizza come:(2x + 3) (4x2 − 6x + 9)(3x − 2) (9x2 + 6x + 4)(3x + 2) (9x2 − 12x + 4)(3x + 2) (9x2 − 6x + 4)(3x − 2) (9x2 + 12x + 4)

To factor the polynomial $27x^3 + 8$, we can recognize that this expression is a sum of cubes, which can be factored using the formula:

$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$

Step 1: Identify $a$ and $b$

In our case, we can identify:

• $a = 3x$ (since $(3x)^3 = 27x^3$)
• $b = 2$ (since $2^3 = 8$)

Step 2: Apply the Formula

Now, we can apply the formula:

$27x^3 + 8 = (3x + 2)((3x)^2 - (3x)(2) + 2^2)$

Step 3: Calculate Each Term

1. Calculate $(3x)^2$: $(3x)^2 = 9x^2$

2. Calculate $- (3x)(2)$: [

   • (3x)(2) = -6x ]

3. Calculate $2^2$: $2^2 = 4$

Step 4: Combine the Terms

Putting it all together, we have: $27x^3 + 8 = (3x + 2)(9x^2 - 6x + 4)$

Final Answer

Thus, the factorization of the polynomial $27x^3 + 8$ is:

$(3x + 2)(9x^2 - 6x + 4)$