Đáp án:
\[\left\{ \begin{array}{l}
a = - 6\\
b = 8\\
c = - 3
\end{array} \right.\]
Giải thích các bước giải:
Ta có:
\(\begin{array}{l}
\left( {{x^4} + a{x^2} + bx + c} \right) \vdots {\left( {x - 1} \right)^3}\\
\Leftrightarrow {x^4} + a{x^2} + bx + c = {\left( {x - 1} \right)^3}\left( {x - d} \right)\\
\Leftrightarrow {x^4} + a{x^2} + bx + c = \left( {{x^3} - 3{x^2} + 3x - 1} \right)\left( {x - d} \right)\\
\Leftrightarrow {x^4} + a{x^2} + bx + c = {x^4} - d{x^3} - 3{x^3} + 3d{x^2} + 3{x^2} - 3dx - x + d\\
\Leftrightarrow {x^4} + a{x^2} + bx + c = {x^4} - \left( {d + 3} \right){x^3} + \left( {3d + 3} \right){x^2} - \left( {3d + 1} \right)x + d\\
\Leftrightarrow \left\{ \begin{array}{l}
- \left( {d + 3} \right) = 0\\
3d + 3 = a\\
- \left( {3d + 1} \right) = b\\
d = c
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
d = - 3\\
a = - 6\\
b = 8\\
c = - 3
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
a = - 6\\
b = 8\\
c = - 3
\end{array} \right.
\end{array}\)
