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Phân tích các đa thức sau thành nhân tử:
a. \({x^3} – 3{x^2} – 4x + 12\)
b. \({x^4} – 5{x^2} + 4\)
c. \({\left( {x + y + z} \right)^3} – {x^3} – {y^3} – {z^3}\)
a. \({x^3} – 3{x^2} – 4x + 12\) \( = \left( {{x^3} – 3{x^2}} \right) – \left( {4x – 12} \right) = {x^2}\left( {x – 3} \right) – 4\left( {x – 3} \right)\)
\( = \left( {x – 3} \right)\left( {{x^2} – 4} \right) = \left( {x – 3} \right)\left( {x + 2} \right)\left( {x – 2} \right)\)
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b. \({x^4} – 5{x^2} + 4\) \( = {x^4} – 4{x^2} – {x^2} + 4 = \left( {{x^4} – 4{x^2}} \right) – \left( {{x^2} – 4} \right)\)
\( = {x^2}\left( {{x^2} – 4} \right) – \left( {{x^2} – 4} \right) = \left( {{x^2} – 4} \right)\left( {{x^2} – 1} \right) = \left( {x + 2} \right)\left( {x – 2} \right)\left( {x + 1} \right)\left( {x – 1} \right)\)
c. \({\left( {x + y + z} \right)^3} – {x^3} – {y^3} – {z^3}\) \( = {\left[ {\left( {x + y} \right) + z} \right]^3} – {x^3} – {y^3} – {z^3}\)
\(\eqalign{ & = {\left( {x + y} \right)^3} + 3{\left( {x + y} \right)^2}z + 3\left( {x + y} \right){z^2} + {z^3} – {x^3} – {y^3} – {z^3} \cr & = {x^3} + {y^3} + 3xy\left( {x + y} \right) + 3{\left( {x + y} \right)^2}z + 3\left( {x + y} \right){z^2} – {x^3} – {y^3} \cr & = 3\left( {x + y} \right)\left[ {xy + \left( {x + y} \right)z + {z^2}} \right] = 3\left( {x + y} \right)\left[ {xy + xz + yz + {z^2}} \right] \cr & = 3\left( {x + y} \right)\left[ {x\left( {y + z} \right) + z\left( {y + z} \right)} \right] = 3\left( {x + y} \right)\left( {y + z} \right)\left( {x + z} \right) \cr} \)