Question

Sa se determine catul si restul pentru impartirea urmatoarelor polinoame:

Answer 1

$\displaystyle a)~f=x^4-3x^3+4x^2-2x+4\\\\g=x^2-2x+1\\\\\left\begin{array}{ccc}~~~~~~~~~~~~~~~~~~x^4-3x^3+4x^2-2x+4\\~-x^4+2x^3-x^2\\~~~~~~~~~~------------- \\~~~~~~~~~~~~~~~/~~-x^3+3x^2-2x+4\\~~~~~~~~~~~~~~~~x^3-2x^2+x\\~~~~~~~~~~-------------\\~~~~~~~~~~~~~~~~~~~~/~~~~~~~x^2-x+4\\~~~~~~~~~~~~~~~~~~~~~~~~~~-x^2+2x-1\\~~~~~~~~~~~~-------------\\~~~~~~~~~~~~~~~~~~~~~~~~~~~~~/~~~~~x+3\end{array}\right| \left\begin{array}{ccc}\cfrac{x^2-2x+1}{x^2-x+1}\\\\\\\\\\\\\\\\\ \end{array}\right\\\\\\ c=x^2-2x+1;~r=x+3$

$\displaystyle b)~f=x^3+2x^2+1\\\\ g=4x^2+3\\\\ \left\begin{array}{ccc}~~~~~~~~~x^3+2x^2+1\\-x^3-\frac{3x}{4}\\~~~~~~~~~--------- \\~~~~~~~~~~~~~~/~~~2x^2-\frac{3x}{4}+1\\~~~~~~~~-2x^2-\frac{3}{2}\\~~~~~~~~~--------- \\~~~~~~~~~~~~~~~~~~/~~~-\frac{3x}{4} -\frac{1}{2} \end{array}\right|\left\begin{array}{ccc}\dfrac{4x^2+3}{\frac{x}{4} +\frac{1}{2} } \\\\\\\\\\\ \end{array}\right\\\\\\ c=\frac{x}{4} +\frac{1}{2} ;~r=-\frac{3x}{4} -\frac{1}{2}$

$c)~f=x^2+2x+2\\\\g=x-2\\\\ \left\begin{array}{ccc}~~~~~~~x^2+2x+2\\-x^2+2x\\~~~-------\\~~~~~~~/~~~~~~4x+2\\~~~~~~~~~~~-4x+8\\~~~-------\\~~~~~~~~~~~~~~~/~~~10\end{array}\right| \left\begin{array}{ccc}\dfrac{x-2}{x+4}\\\\\\\\\\\ \end{array}\right\\\\\\c=x+4;~r=10$