Bài 1. Rút gọn biểu thức: \(A = \left( {m – n} \right)\left( {{m^2} + mn + {n^2}} \right) – \left( {m + n} \right)\left( {{m^2} – mn + {n^2}} \right).\)
Bài 2. Chứng minh rằng: \(\left( {a – 1} \right)\left( {a – 2} \right)\left( {1 + a + {a^2}} \right)\left( {4 + 2a + {a^2}} \right) = {a^6} – 9{a^3} + 8.\)
Bài 3. Tìm x, biết: \(\left( {x + 2} \right)\left( {{x^2} – 2x + 4} \right) – x\left( {x – 3} \right)\left( {x + 3} \right) = 26.\)
Bài 1. Ta có:
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\(A = \left( {{m^3} – {n^3}} \right) – \left( {{m^3} + {n^3}} \right) = – 2{n^3}.\)
Bài 2. Ta có:
\(\left( {a – 1} \right)\left( {a – 2} \right)\left( {1 + a + {a^2}} \right)\left( {4 + 2a + {a^2}} \right)\)
\( = \left( {a – 1} \right)\left( {{a^2} + a + 1} \right)\left( {a – 2} \right)\left( {{a^2} + 2a + 4} \right)\)
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\( = \left( {{a^3} – 1} \right)\left( {{a^3} – 8} \right) \)
\(= {a^6} – 8{a^3} – {a^3} + 8 \)
\(= {a^6} – 9{a^3} + 8\) (đpcm).
Bài 3. Ta có:
\(\left( {x + 2} \right)\left( {{x^2} – 2x + 4} \right) – x\left( {x – 3} \right)\left( {x + 3} \right)\)
\( = {x^3} + 8 – x\left( {{x^2} – 9} \right) \)
\(= {x^3} + 8 – {x^3} + 9x = 9x + 8\)
Vậy \(9x + 8 = 26 \Rightarrow x = 2.\)
