Câu hỏi 1.23 - Mục Bài tập trang 14

1. Nội dung câu hỏi

Rút gọn biểu thức:

a) \(\left( {x - y} \right)\left( {y + z} \right)\left( {z + x} \right) + \left( {x + y} \right)\left( {y - z} \right)\left( {z + x} \right) + \left( {x + y} \right)\left( {y + z} \right)\left( {z - x} \right)\);

b) \(\left( {2x + y} \right)\left( {2y + z} \right)\left( {2z + x} \right) - \left( {2x - y} \right)\left( {2y - z} \right)\left( {2z - x} \right)\).

3. Lời giải chi tiết

Đặt \(A = \left( {x - y} \right)\left( {y + z} \right)\left( {z + x} \right)\); \(B = \left( {x + y} \right)\left( {y - z} \right)\left( {z + x} \right)\); \(C = \left( {x + y} \right)\left( {y + z} \right)\left( {z - x} \right)\).

Ta xét:

\(A = \left( {x - y} \right)\left( {y + z} \right)\left( {z + x} \right)\)

\( = \left( {xy + xz - {y^2} - yz} \right)\left( {z + x} \right)\)

\( = xyz + {x^2}y + x{z^2} + {x^2}z - {y^2}z - x{y^2} - y{z^2} - xyz\)

\( = \left( {xyz - xyz} \right) + {x^2}y + x{z^2} + {x^2}z - {y^2}z - x{y^2} - y{z^2}\)

\( = {x^2}y + x{z^2} + {x^2}z - {y^2}z - x{y^2} - y{z^2}\).

Tương tự

\(B = \left( {x + y} \right)\left( {y - z} \right)\left( {z + x} \right)\)

\( = \left( {xy - xz + {y^2} - yz} \right)\left( {z + x} \right)\)

\( = xyz + {x^2}y - x{z^2} - {x^2}z + {y^2}z + x{y^2} - y{z^2} - xyz\)

\( = {x^2}y - x{z^2} - {x^2}z + {y^2}z + x{y^2} - y{z^2}\).

\(C = \left( {x + y} \right)\left( {y + z} \right)\left( {z - x} \right)\)

\( = \left( {xy + xz + {y^2} + yz} \right)\left( {z - x} \right)\)

\( = xyz - {x^2}y + x{z^2} - {x^2}z + {y^2}z - x{y^2} + y{z^2} - xyz\)

\( = \left( {xyz - xyz} \right) - {x^2}y + x{z^2} - {x^2}z + {y^2}z - x{y^2} + y{z^2}\)

\( =  - {x^2}y + x{z^2} - {x^2}z + {y^2}z - x{y^2} + y{z^2}\).

Khi đó

\(\left( {x - y} \right)\left( {y + z} \right)\left( {z + x} \right) + \left( {x + y} \right)\left( {y - z} \right)\left( {z + x} \right) + \left( {x + y} \right)\left( {y + z} \right)\left( {z - x} \right) = A + B + C\)

\(\begin{array}{l} = {x^2}y + x{z^2} + {x^2}z - {y^2}z - x{y^2} - y{z^2} + {x^2}y - x{z^2} - {x^2}z + {y^2}z + x{y^2} - y{z^2} + \\ - {x^2}y + x{z^2} - {x^2}z + {y^2}z - x{y^2} + y{z^2}\end{array}\)

\(\begin{array}{l} = \left( {{x^2}y + {x^2}y - {x^2}y} \right) + \left( { - x{y^2} + x{y^2} - x{y^2}} \right) + \left( {x{z^2} - x{z^2} + x{z^2}} \right) + \left( {{x^2}z - {x^2}z - {x^2}z} \right)\\ + \left( { - {y^2}z + {y^2}z + {y^2}z} \right) + \left( { - y{z^2} - y{z^2} + y{z^2}} \right)\end{array}\)

\( = {x^2}y - x{y^2} + x{z^2} - {x^2}z + {y^2}z - y{z^2}\).

b)

Đặt \(M = \left( {2x + y} \right)\left( {2y + z} \right)\left( {2z + x} \right)\); \(N = \left( {2x - y} \right)\left( {2y - z} \right)\left( {2z - x} \right)\).

Ta xét

\(M = \left( {2x + y} \right)\left( {2y + z} \right)\left( {2z + x} \right)\)

\( = \left( {4xy + 2xz + 2{y^2} + yz} \right)\left( {2z + x} \right)\)

\( = 8xyz + 4{x^2}y + 4x{z^2} + 2{x^2}z + 4{y^2}z + 2x{y^2} + 2y{z^2} + xyz\)

\( = \left( {8xyz + xyz} \right) + 4{x^2}y + 4x{z^2} + 2{x^2}z + 4{y^2}z + 2x{y^2} + 2y{z^2}\)

\( = 9xyz + 4{x^2}y + 4x{z^2} + 2{x^2}z + 4{y^2}z + 2x{y^2} + 2y{z^2}\)

Tương tự

\(N = \left( {2x - y} \right)\left( {2y - z} \right)\left( {2z - x} \right)\)

\( = \left( {4xy - 2xz - 2{y^2} + yz} \right)\left( {2z - x} \right)\)

\( = 8xyz - 4{x^2}y - 4x{z^2} + 2{x^2}z - 4{y^2}z + 2x{y^2} + 2y{z^2} - xyz\)

\( = \left( {8xyz - xyz} \right) - 4{x^2}y - 4x{z^2} + 2{x^2}z - 4{y^2}z + 2x{y^2} + 2y{z^2}\)

\( = 7xyz - 4{x^2}y - 4x{z^2} + 2{x^2}z - 4{y^2}z + 2x{y^2} + 2y{z^2}\).

Do đó

\(\left( {2x + y} \right)\left( {2y + z} \right)\left( {2z + x} \right) - \left( {2x - y} \right)\left( {2y - z} \right)\left( {2z - x} \right) = M - N\)

\(\begin{array}{l} = \left( {9xyz + 4{x^2}y + 4x{z^2} + 2{x^2}z + 4{y^2}z + 2x{y^2} + 2y{z^2}} \right)\\ - \left( {7xyz - 4{x^2}y - 4x{z^2} + 2{x^2}z - 4{y^2}z + 2x{y^2} + 2y{z^2}} \right)\end{array}\)
\(\begin{array}{l} = 9xyz + 4{x^2}y + 4x{z^2} + 2{x^2}z + 4{y^2}z + 2x{y^2} + 2y{z^2} - 7xyz + \\ + 4{x^2}y + 4x{z^2} - 2{x^2}z + 4{y^2}z - 2x{y^2} - 2y{z^2}\end{array}\)

\( = \left( {9xyz - 7xyz} \right) + \left( {4{x^2}y + 4{x^2}y} \right) + \left( {4{y^2}z + 4{y^2}z} \right) + \left( {4x{z^2} + 4x{z^2}} \right) + \)

\( + \left( {2x{y^2} - 2x{y^2}} \right) + \left( {2y{z^2} - 2y{z^2}} \right) + \left( {2{x^2}z - 2{x^2}z} \right)\)

\( = 2xyz + 8{x^2}y + 8{y^2}z + 8x{z^2}\).