cho đa thức:
f(x) = 2x^2011 – 3x^2010 + 2x^2009 – 3x^2008 + … + 2x^3 – 3x^2 + 2x – 3.
Hãy tính f(a) biết 2a^2 = 3a.
cho đa thức:
f(x) = 2x^2011 – 3x^2010 + 2x^2009 – 3x^2008 + … + 2x^3 – 3x^2 + 2x – 3.
Hãy tính f(a) biết 2a^2 = 3a.
Đáp án:
$\begin{array}{l}
f\left( x \right) = 2{x^{2011}} – 3{x^{2010}} + 2{x^{2009}} – 3{x^{2008}}\\
+ … + 2{x^3} – 3{x^2} + 2x – 3\\
= {x^{2010}}\left( {2x – 3} \right) + {x^{2008}}\left( {2x – 3} \right) + … + {x^2}\left( {2x – 3} \right) + \left( {2x – 3} \right)\\
= \left( {2x – 3} \right)\left( {{x^{2010}} + {x^{2008}} + … + {x^2} + 1} \right)\\
Do:2{a^2} = 3a\\
\Rightarrow a.\left( {2a – 3} \right) = 0\\
\Rightarrow \left[ \begin{array}{l}
a = 0\\
2a – 3 = 0
\end{array} \right.\\
+ Khi:a = 0\\
\Rightarrow f\left( a \right) = \left( {2a – 3} \right).\left( {{a^{2010}} + {a^{2008}} + … + {a^2} + 1} \right)\\
= \left( { – 3} \right).1 = – 3\\
+ Khi:2a – 3 = 0\\
\Rightarrow f\left( a \right) = 0\\
\text{Vậy}\,f\left( a \right) = – 3\,\text{hoặc}\,f\left( a \right) = 0
\end{array}$