%5E3%2B2x%5E2%2B4xy%2B2y%5E2.jpeg "(x+y)^3+2x^2+4xy+2y^2")
Simplifying the Expression: (x + y)³ + 2x² + 4xy + 2y²
This article aims to simplify the given expression: (x + y)³ + 2x² + 4xy + 2y²
Expanding the Cube
First, we need to expand the cube of the binomial (x + y):
(x + y)³ = (x + y)(x + y)(x + y)
We can use the distributive property (or FOIL method) to expand this:
- (x + y)(x + y) = x² + 2xy + y²
- (x² + 2xy + y²)(x + y) = x³ + 3x²y + 3xy² + y³
Therefore, (x + y)³ = x³ + 3x²y + 3xy² + y³
Combining Terms
Now, let's substitute this back into our original expression:
(x + y)³ + 2x² + 4xy + 2y² = x³ + 3x²y + 3xy² + y³ + 2x² + 4xy + 2y²
Next, we combine like terms:
- x³ remains the same.
- 3x²y + 2x² = 5x²y
- 3xy² + 4xy = 7xy²
- y³ + 2y² = y³ + 2y²
Simplified Expression
Finally, we arrive at the simplified form of the expression:
(x + y)³ + 2x² + 4xy + 2y² = x³ + 5x²y + 7xy² + y³ + 2y²
This expression cannot be simplified further as there are no more like terms.