CMR: x^4-6x^3+27x^2-54x+32 chia hết cho 2 27/09/2021 Bởi aihong CMR: x^4-6x^3+27x^2-54x+32 chia hết cho 2
\[\begin{array}{l} {x^4} – 6{x^3} + 27{x^2} – 54x + 32\\ = {x^4} – {x^3} – 5{x^3} + 5{x^2} + 22{x^2} – 22x – 32x + 32\\ = {x^3}\left( {x – 1} \right) – 5{x^2}\left( {x – 1} \right) + 22x\left( {x – 1} \right) – 32\left( {x – 1} \right)\\ = \left( {x – 1} \right)\left( {{x^3} – 5{x^2} + 22x – 32} \right)\\ = \left( {x – 1} \right)\left( {{x^3} – 2{x^2} – 3{x^2} + 6x + 16x – 32} \right)\\ = \left( {x – 1} \right)\left[ {{x^2}\left( {x – 2} \right) – 3x\left( {x – 2} \right) + 16\left( {x – 2} \right)} \right]\\ = \left( {x – 1} \right)\left( {x – 2} \right)\left( {{x^2} – 3x + 16} \right). \end{array}\] Bình luận
\[\begin{array}{l}
{x^4} – 6{x^3} + 27{x^2} – 54x + 32\\
= {x^4} – {x^3} – 5{x^3} + 5{x^2} + 22{x^2} – 22x – 32x + 32\\
= {x^3}\left( {x – 1} \right) – 5{x^2}\left( {x – 1} \right) + 22x\left( {x – 1} \right) – 32\left( {x – 1} \right)\\
= \left( {x – 1} \right)\left( {{x^3} – 5{x^2} + 22x – 32} \right)\\
= \left( {x – 1} \right)\left( {{x^3} – 2{x^2} – 3{x^2} + 6x + 16x – 32} \right)\\
= \left( {x – 1} \right)\left[ {{x^2}\left( {x – 2} \right) – 3x\left( {x – 2} \right) + 16\left( {x – 2} \right)} \right]\\
= \left( {x – 1} \right)\left( {x – 2} \right)\left( {{x^2} – 3x + 16} \right).
\end{array}\]