Phân tích mỗi đa thức sau thành nhân tử: a) \(8{x^3}yz + 12{x^2}yz + 6xyz + yz\); b) \(81{x^4}\left( {{z^2} - {y^2}} \right) - {z^2} + {y^2}\); c) \({x^4} - 5{x^2} + 4\).

a) \(8{x^3}yz + 12{x^2}yz + 6xyz + yz\)

\( = yz\left( {8{x^3} + 12{x^2} + 6x + 1} \right)\)

\( = yz\left[ {{{\left( {2x} \right)}^3} + 3 \cdot {{\left( {2x} \right)}^2} \cdot 1 + 3 \cdot 2x \cdot {1^2} + {1^3}} \right]\)

\( = yz{\left( {2x + 1} \right)^3}\).

b) \(81{x^4}\left( {{z^2} - {y^2}} \right) - {z^2} + {y^2}\)

\( = 81{x^4}\left( {{z^2} - {y^2}} \right) - \left( {{z^2} - {y^2}} \right)\)

\( = \left( {{z^2} - {y^2}} \right)\left( {81{x^4} - 1} \right)\)

\( = \left( {z - y} \right)\left( {z + y} \right)\left[ {{{\left( {9{x^2}} \right)}^2} - {1^2}} \right]\)

\( = \left( {z - y} \right)\left( {z + y} \right)\left( {9{x^2} + 1} \right)\left( {9{x^2} - 1} \right)\)

\( = \left( {z - y} \right)\left( {z + y} \right)\left( {9{x^2} + 1} \right)\left[ {{{\left( {3x} \right)}^2} - {1^2}} \right]\)

\( = \left( {z - y} \right)\left( {z + y} \right)\left( {9{x^2} + 1} \right)\left( {3x + 1} \right)\left( {3x - 1} \right)\).

c) \({x^4} - 5{x^2} + 4\)

\( = {x^4} - {x^2} - 4{x^2} + 4\)

\( = {x^2}\left( {{x^2} - 1} \right) - 4\left( {{x^2} - 1} \right)\)

\( = \left( {{x^2} - 4} \right)\left( {{x^2} - 1} \right)\)

\( = \left( {x - 2} \right)\left( {x + 2} \right)\left( {x - 1} \right)\left( {x + 1} \right).\)