Đáp án:
a. \(A\left( x \right) = {x^3} - \frac{5}{2}{x^2} - \frac{9}{2}x + 11\)
Giải thích các bước giải:
\(\begin{array}{l}
M\left( x \right) = - {x^3} + \frac{5}{2}{x^2} - \frac{1}{2}x - 1\\
N\left( x \right) = - 2{x^3} + 5{x^2} + 4x - 12\\
a.A\left( x \right) = M\left( x \right) - N\left( x \right)\\
= - {x^3} + \frac{5}{2}{x^2} - \frac{1}{2}x - 1 + 2{x^3} - 5{x^2} - 4x + 12\\
= {x^3} - \frac{5}{2}{x^2} - \frac{9}{2}x + 11\\
A\left( x \right) = 0\\
\to {x^3} - \frac{5}{2}{x^2} - \frac{9}{2}x + 11 = 0\\
\to 2{x^3} - 5{x^2} - 9x + 22 = 0\\
\to 2{x^3} - 4{x^2} - {x^2} + 2x - 11x + 22 = 0\\
\to 2{x^2}\left( {x - 2} \right) - x\left( {x - 2} \right) - 11\left( {x - 2} \right) = 0\\
\to \left( {x - 2} \right)\left( {2{x^2} - x - 11} \right) = 0\\
\to \left[ \begin{array}{l}
x = 2\\
2{x^2} - x - 11 = 0
\end{array} \right.\\
b.B\left( x \right) = M\left( x \right) + N\left( x \right)\\
= - {x^3} + \frac{5}{2}{x^2} - \frac{1}{2}x - 1 - 2{x^3} + 5{x^2} + 4x - 12\\
= - 3{x^3} + \frac{{15}}{2}{x^2} + \frac{7}{2}x - 13\\
\to Bậc = 3
\end{array}\)
