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Phân tích các đa thức sau thành nhân tử:
a)\({a^2} – {b^2} – 4a + 4;\)
b) \({x^2} + 2x – 3\)
c) \(4{x^2}{y^2} – {\left( {{x^2}{y^2}} \right)^2}\)
d) \(2{a^3} – 54{b^3}\) .
Hướng dẫn làm bài:
a) \({a^2} – {b^2} – 4a + 4 \Leftrightarrow {a^2} – 4a + 4 – {b^2}\)
= \({\left( {a – 2} \right)^2} – {b^2} = \left( {a – 2 + b} \right)\left( {a – 2 – b} \right)\)
= \(\left( {a + b – 2} \right)\left( {a – b – 2} \right)\)
b) \({x^2} + 2x – 3 = {x^2} + 2x + 1 – 4\)
=\({\left( {x + 1} \right)^2} – {2^2} = \left( {x + 1 + 2} \right)\left( {x + 1 – 2} \right)\)
=\(\left( {x + 3} \right)\left( {x – 1} \right)\)
c) \(4{x^2}{y^2} – {\left( {{x^2}{y^2}} \right)^2} = {\left( {2xy} \right)^2} – {\left( {{x^2} + {y^2}} \right)^2}\)
= \(\left( {2xy – {x^2} – {y^2}} \right)\left( {2xy + {x^2} + {y^2}} \right)\)
=\( – \left( {{x^2} – 2xy + {y^2}} \right)\left( {{x^2} + 2xy + {y^2}} \right)\)
=\( – {\left( {x – y} \right)^2}{\left( {x + y} \right)^2}\)
d) \(2{a^3} – 54{b^3} = 2\left( {{a^3} – 27{b^3}} \right)\)
=\(2\left[ {{a^3} – {{\left( {3b} \right)}^3}} \right] = 2\left( {a – 3b} \right)\left( {{a^2} + 3ab + 9{b^2}} \right)\).