To match each polynomial with its appropriately factored expression, we will analyze and detail each polynomial factorization one by one.
### Step-by-Step Factorization Analysis:
#### 1. Factorizing [tex]\(8x^3 + 27\)[/tex]:
Given polynomial: [tex]\(8x^3 + 27\)[/tex]
This polynomial is a sum of cubes. The standard formula for a sum of cubes [tex]\(a^3 + b^3\)[/tex] is:
[tex]\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \][/tex]
Here, [tex]\(a = 2x\)[/tex] and [tex]\(b = 3\)[/tex].
Applying the sum of cubes formula, we get:
[tex]\[ (2x)^3 + 3^3 = (2x + 3)((2x)^2 - 2x \cdot 3 + 3^2) \][/tex]
Simplifying the parts inside the parentheses:
[tex]\[ (2x + 3)(4x^2 - 6x + 9) \][/tex]
So, the factored form of [tex]\(8x^3 + 27\)[/tex] is:
[tex]\[ (2x + 3)(4x^2 - 6x + 9) \][/tex]
#### 2. Factorizing [tex]\(8x^3 - 27\)[/tex]:
Given polynomial: [tex]\(8x^3 - 27\)[/tex]
This polynomial is a difference of cubes. The standard formula for a difference of cubes [tex]\(a^3 - b^3\)[/tex] is:
[tex]\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \][/tex]
Here, [tex]\(a = 2x\)[/tex] and [tex]\(b = 3\)[/tex].
Applying the difference of cubes formula, we get:
[tex]\[ (2x)^3 - 3^3 = (2x - 3)((2x)^2 + 2x \cdot 3 + 3^2) \][/tex]
Simplifying the parts inside the parentheses:
[tex]\[ (2x - 3)(4x^2 + 6x + 9) \][/tex]
So, the factored form of [tex]\(8x^3 - 27\)[/tex] is:
[tex]\[ (2x - 3)(4x^2 + 6x + 9) \][/tex]
### Matching the Polynomials with Their Factored Forms:
- [tex]\(8x^3 + 27\)[/tex] matches with [tex]\((2x + 3)(4x^2 - 6x + 9)\)[/tex]
- [tex]\(8x^3 - 27\)[/tex] matches with [tex]\((2x - 3)(4x^2 + 6x + 9)\)[/tex]
In summary:
- [tex]\(8 x^3 + 27\)[/tex] → [tex]\((2 x + 3)(4 x^2 - 6 x + 9)\)[/tex]
- [tex]\(8 x^3 - 27\)[/tex] → [tex]\((2 x - 3)(4 x^2 + 6 x + 9)\)[/tex]
These are the correct factorizations for each polynomial.
