phân tích các đa thức sau thành nhân tử (x-4)(x-5)(x-6)(x-7)-1680

Đáp án:

$\begin{array}{l}
A = \left( {x – 4} \right)\left( {x – 5} \right)\left( {x – 6} \right)\left( {x – 7} \right) – 1680\\
 = \left( {x – 4} \right)\left( {x – 7} \right)\left( {x – 5} \right)\left( {x – 6} \right) – 1680\\
 = \left( {{x^2} – 7x – 4x + 28} \right)\left( {{x^2} – 6x – 5x + 30} \right) – 1680\\
 = \left( {{x^2} – 11x + 28} \right)\left( {{x^2} – 11x + 30} \right) – 1680\\
Dat:{x^2} – 11x + 28 = a\\
 \Rightarrow {x^2} – 11x + 30 = a + 2\\
 \Rightarrow A = a.\left( {a + 2} \right) – 1680\\
 = {a^2} + 2a – 1680\\
 = {a^2} + 42a – 40a – 1680\\
 = \left( {a + 42} \right)\left( {a – 40} \right)\\
 = \left( {{x^2} – 11x + 28 + 42} \right).\left( {{x^2} – 11x + 28 – 40} \right)\\
 = \left( {{x^2} – 11x + 70} \right)\left( {{x^2} – 11x – 12} \right)
\end{array}$