%2B3(x%5E2%2B4x%2B4)y%2B3(x%2B2)y%5E2%2By%5E3.jpeg "(x^3+6x^2+12x+8)+3(x^2+4x+4)y+3(x+2)y^2+y^3")
Factoring the Expression: (x^3+6x^2+12x+8)+3(x^2+4x+4)y+3(x+2)y^2+y^3
This expression looks complex, but we can simplify it by recognizing a pattern and using factoring techniques.
Recognizing the Pattern:
Observe that the coefficients in the expression follow a pattern similar to the binomial expansion:
- (x + a)^3 = x^3 + 3ax^2 + 3a^2x + a^3
Let's try to fit our expression into this pattern.
- x^3 + 6x^2 + 12x + 8 can be rewritten as (x + 2)^3
- 3(x^2 + 4x + 4)y can be rewritten as 3(x + 2)^2 y
- 3(x + 2)y^2 remains the same
- y^3 remains the same
Factoring the Expression:
Now, we can rewrite the entire expression as:
(x + 2)^3 + 3(x + 2)^2 y + 3(x + 2)y^2 + y^3
This expression follows the pattern of the binomial expansion for (a + b)^3:
(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
Therefore, our expression can be factored as:
(x + 2 + y)^3
Final Answer:
The factored form of the given expression is (x + 2 + y)^3.