✰Answer:
6x(9x² - y²) + 6y²
✰Step-by-step explanation:
To factorize the expression (3x + y)³ + (3x - y)³, we can use the formula for the sum of cubes:
a³ + b³ = (a + b)(a² - ab + b²)
Applying this formula to our expression, we get:
(3x + y)³ + (3x - y)³ = [(3x + y) + (3x - y)][(3x + y)² - (3x + y)(3x - y) + (3x - y)²]
Simplifying further, we have:
(3x + y)³ + (3x - y)³ = [6x][(3x + y)² - (9x² - y²) + (3x - y)²]
Now, let's expand and simplify the terms inside the square brackets:
(3x + y)² = (3x + y)(3x + y) = 9x² + 6xy + y²
(3x - y)² = (3x - y)(3x - y) = 9x² - 6xy + y²
(3x + y)(3x - y) = 9x² - y²
Substituting these values back into our expression, we get:
(3x + y)³ + (3x - y)³ = [6x][(9x² + 6xy + y²) - (9x² - y²) + (9x² - 6xy + y²)]
Simplifying the terms inside the square brackets:
(9x² + 6xy + y²) - (9x² - y²) + (9x² - 6xy + y²) = 9x² + 6xy + y² - 9x² + y² + 9x² - 6xy + y²
Combining like terms:
9x² - 9x² + 9x² + 6xy - 6xy + y² + y² + y² = 3y² + 3y²
Simplifying further:
3y² + 3y² = 6y²
Therefore, the factorized form of (3x + y)³ + (3x - y)³ is:
6x(9x² - y²) + 6y²
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