Phân tích đa thức thành nhân tử:

A = 8(a + b + c)³ - (2a + b - c)³ - (2b + c - a)³ - (2c + a - b)³

Lời giải:

Đặt \(\left\{ \begin{array}{l}x = 2a + b - c\\y = 2b + c - a\\z = 2c + a - b\end{array} \right.\)

⇒ x + y + z = 2a + b – c + 2b + c – a + 2c + a – b = 2a + 2b + 2c = 2(a + b + c)

A = 8(a + b + c)³ - (2a + b - c)³ - (2b + c - a)³ - (2c + a - b)³

= (x + y + z)³ − x³ − y³ − z³

= (x + y + z − x)[(x + y + z)² + x(x + y + z ) + x²] − (y + z)(y² – yz + z²)

= (y + z)(x² + y² + z² + 2xy + 2yz + 2xz + x² + xy + xz + x² − y² + yz − z²)

= (y + z)(3x² + 3xy + 3yz + 3xz)

= 3(x + y)(y + z)(x + z)

= 3(2a + b – c + 2b + c − a)(2b + c – a + 2c + a − b)(2a + b – c + 2c + a − b)

=3(a + 3b)(b + 3c)(c + 3a)