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