%5E3-27.jpeg "(3x-2)^3-27")
Factoring (3x-2)^3 - 27
This expression is a difference of cubes, a common pattern in algebra that can be factored. Let's break down the steps:
1. Recognizing the Pattern
The expression (3x-2)^3 - 27 can be rewritten as:
- (3x-2)^3 - 3^3
This form clearly shows the difference of cubes pattern, where:
- a = (3x-2)
- b = 3
2. Applying the Difference of Cubes Formula
The difference of cubes formula states:
- a^3 - b^3 = (a - b)(a^2 + ab + b^2)
Applying this to our expression:
- [(3x-2) - 3][(3x-2)^2 + (3x-2)(3) + 3^2]
3. Simplifying the Expression
Let's simplify each part of the factored expression:
- (3x - 5)
- (9x^2 - 12x + 4) + (9x - 6) + 9
4. Final Factored Form
Combining the simplified terms, we get the fully factored expression:
- (3x - 5)(9x^2 - 3x + 7)
Therefore, the factored form of (3x-2)^3 - 27 is (3x - 5)(9x^2 - 3x + 7).