The expression (a - b)³ - 8·(a + b)³ can be factored using the formula for the difference of two cubes to get;
(a - b)³ + 8·(a + b)³ = (-a - 3·b)·((a - b)² + 2·(a² - b²) + 4·(a + b)²)
The steps used in factoring the expression can be presented as follows;
The expression (a - b)³ - 8·(a + b)³ can be factored using the formula for the difference of two cubes as follows;
x³ - y³ = (x - y) × (x² + x·y + y²)
The expression 8·(a + b)³ can be presented as; 8·(a + b)³ = (2·(a + b))³
Therefore; (a - b)³ - (2·(a + b))³
(a - b)³ - (2·(a + b))³ = ((a - b) - (2·(a + b))) × ((a - b)² + (a - b)·(2·(a + b)) + (2·(a + b))²)
((a - b) - (2·(a + b))) = (-a - 3·b)
(a - b)·(2·(a + b)) = 2·(a² - b²)
(2·(a + b))² = 4·(a + b)²
((a - b) - (2·(a + b))) × ((a - b)² + (a - b)·(2·(a + b)) + (2·(a + b))²) = (-a - 3·b)·((a - b)² + 2·(a² - b²) + 4·(a + b)²)
The complete question obtained from a similar question found through search can be presented as follows;
Factorize each of the following expressions;
(a - b)³ - 8·(a + b)³