Ejercicios Resueltos de Polinomios

Dados
$p(x)=x^4-2x+1$ ; $q(x)=-x^3-4x^2-3$ ; $n(x)=x^2+4x$

Hallar:
a)  $\displaystyle\frac{p(x)}{q(x)}$

      $x^4 - 2x + 1 \,\,\,\,\,\,│\underline{-x^3 - 4x^2 - 3}$
 $\underline{-x^4-4x^3-3x}$        $\color{red}{-x+4}$ $ \,\,\, \mbox{polinomio }cociente\color{red}\bullet$
           $-4x^3-5x+1$
               $\underline{4x^3+16x^2+12}$
                       $\color{red}{16x^2-5x+13}$   $ \,\,\, \mbox{polinomio }residuo\color{red}\bullet$

b)  $p(x)-[n(x)+q(x)]$
    $x^4-2x+1-[(x^2+4x)+(-x^3-4x^2-3)]$
→    $x^4-2x+1-x^2-4x+x^3+4x^2+3$
→    $x^4+x^3+3x^2-6x+4\color{red}\bullet$

c)  $\displaystyle\frac{p(x)+[n(x)]^2}{q(x)}$

      $x^4 - 2x + 1+(x^2+4x)^2 \,\,\,\,\,\,│\underline{-x^3 - 4x^2 - 3}$
      $x^4 - 2x + 1+x^4 + 8x^3 + 16x^2 \,\,\,\,\,\,│\underline{-x^3 - 4x^2 - 3}$
      $2x^4 + 8x^3 + 16x^2 - 2x + 1 \,\,\,\,\,\,│\underline{-x^3 - 4x^2 - 3}$
  $\underline{-2x^4-8x^3-6x}$  $\color{red}{-2x}$ $ \,\,\, \mbox{polinomio }cociente\color{red}\bullet$
                       $\color{red}{16x^2 - 8x + 1}$   $ \,\,\, \mbox{polinomio }residuo\color{red}\bullet$