Phân tích các đa thức sau thành nhân tử:
a) \(m{x^2} + m{y^2} – n{x^2} – n{y^2}\)
b) \(40bc + 9cx – 24bx – 15{c^2}\)
c) \(a\left( {{b^2} + {c^2} – {a^2}} \right) + b\left( {{c^2} + {a^2} – {b^2}} \right).\)
a) \(m{x^2} + m{y^2} – n{x^2} – n{y^2} \)
\(= m\left( {{x^2} + {y^2}} \right) – n\left( {{x^2} + {y^2}} \right) \)
\(= \left( {{x^2} + {y^2}} \right)\left( {m – n} \right).\)
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b) \(40bc + 9cx – 24bx – 15{c^2}\)
\(= \left( {40bc – 15{c^2}} \right) + \left( {9cx – 24bx} \right)\)
\( = 5c\left( {8b – 3c} \right) + 3x\left( {3c – 8b} \right) \)
\( = \left( {8b – 3c} \right)\left( {5c – 3x.} \right)\)
c) \(a\left( {{b^2} + {c^2} – {a^2}} \right) + b\left( {{c^2} + {a^2} – {b^2}} \right) \)
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\(= a{b^2} + a{c^2} – {a^3} + b{c^2} + b{a^2} – {b^3}\)
\( = \left( {a{b^2} + b{a^2}} \right) + \left( {a{c^2} + b{c^2}} \right) – \left( {{a^3} + {b^3}} \right)\)
\( = ab\left( {a + b} \right) + {c^2}\left( {a + b} \right) – \left( {a + b} \right)\left( {{a^2} – ab + {b^2}} \right)\)
\( = \left( {a + b} \right)\left( {ab + {c^2} – {a^2} + ab – {b^2}} \right) \)
\(= \left( {a + b} \right)\left[ {{c^2} + \left( { – {a^2} + 2ab – {b^2}} \right)} \right]\)
\( = \left( {a + b} \right)\left[ {{c^2} – {{\left( {a – b} \right)}^2}} \right] \)
\(= \left( {a + b} \right)\left( {c + a – b} \right)\left( {c – a + b} \right).\)
