Ejercicios sección 3.3

En los ejercicios 1 al 10, use la división larga entre los polinomios indicados para determinar el cociente y el residuo.

  1. f(x)=5x2+2x−3;    g(x)=x2+1\hspace{0.1cm}f_{\left(x\right)}=5x^2+2x-3;\;\; g_{\left(x\right)}=x^2+1\\[0.2cm]
  2. f(x)=14x3−12x2+6;    g(x)=x2−1\hspace{0.1cm}f_{\left(x\right)}=14x^3-12x^2+6;\;\; g_{\left(x\right)}=x^2-1\\[0.2cm]
  3. f(x)=x3+x2+x+1;    g(x)=(2x+1)2\hspace{0.1cm}f_{\left(x\right)}=x^3+x^2+x+1;\;\; g_{\left(x\right)}=\left(2x+1\right)^2\\[0.2cm]
  4. f(x)=x4+8;    g(x)=x3+2x−1\hspace{0.1cm}f_{\left(x\right)}=x^4+8;\;\; g_{\left(x\right)}=x^3+2x-1\\[0.2cm]
  5. f(x)=5x6−x5+10x4+3x2−2x+4;  g(x)=x2+x−1\hspace{0.1cm}f_{\left(x\right)}=5x^6-x^5+10x^4+3x^2-2x+4;\\\; g_{\left(x\right)}=x^2+x-1\\[0.2cm]
  1. f(x)=2x4−x3−3x2+7x−12;    g(x)=x2−3\hspace{0.1cm}f_{\left(x\right)}=2x^4-x^3-3x^2+7x-12;\;\; g_{\left(x\right)}=x^2-3\\[0.2cm]
  2. f(x)=3x4+2x3−x2−x−6;    g(x)=x2+1\hspace{0.1cm}f_{\left(x\right)}=3x^4+2x^3-x^2-x-6;\;\; g_{\left(x\right)}=x^2+1\\[0.2cm]
  3. f(x)=3x3−5x2−4x−8;    g(x)=2x2+x\hspace{0.1cm}f_{\left(x\right)}=3x^3-5x^2-4x-8;\;\; g_{\left(x\right)}=2x^2+x\\[0.2cm]
  4. f(x)=x3−3x+9;    g(x)=−5x2+3\hspace{0.1cm}f_{\left(x\right)}=x^3-3x+9;\;\; g_{\left(x\right)}=-5x^2+3\\[0.2cm]
  5. f(x)=7x2+3x+10;    g(x)=x2−x+10\hspace{0.1cm}f_{\left(x\right)}=7x^2+3x+10;\;\; g_{\left(x\right)}=x^2-x+10\\[0.2cm]

En los ejercicios 11 al 20, use la división sintética (Ruffini) entre los polinomios indicados para determinar el cociente y el residuo.

  1. f(x)=2x3−3x2+4x−5;    g(x)=x−2\hspace{0.1cm}f_{\left(x\right)}=2x^3-3x^2+4x-5;\;\; g_{\left(x\right)}=x-2\\[0.2cm]
  2. f(x)=3x3−4x2−x+8;    g(x)=x+4\hspace{0.1cm}f_{\left(x\right)}=3x^3-4x^2-x+8;\;\; g_{\left(x\right)}=x+4\\[0.2cm]
  3. f(x)=5x3−6x2+15;    g(x)=x−4\hspace{0.1cm}f_{\left(x\right)}=5x^3-6x^2+15;\;\; g_{\left(x\right)}=x-4\\[0.2cm]
  4. f(x)=−2x4+10x−3;    g(x)=x−3\hspace{0.1cm}f_{\left(x\right)}=-2x^4+10x-3;\;\; g_{\left(x\right)}=x-3\\[0.2cm]
  5. f(x)=9x3−6x2+3x−4;  g(x)=x−13\hspace{0.1cm}f_{\left(x\right)}=9x^3-6x^2+3x-4;\\\; g_{\left(x\right)}=x-\frac{1}{3}\\[0.2cm]
  1. f(x)=4x2−8x+6;    g(x)=x−12\hspace{0.1cm}f_{\left(x\right)}=4x^2-8x+6;\;\; g_{\left(x\right)}=x-\frac{1}{2}\\[0.2cm]
  2. f(x)=4x3−3x2+2x+4;    g(x)=x−7\hspace{0.1cm}f_{\left(x\right)}=4x^3-3x^2+2x+4;\;\; g_{\left(x\right)}=x-7\\[0.2cm]
  3. f(x)=4x4+3x3−x2−5x−6;    g(x)=x+3\hspace{0.1cm}f_{\left(x\right)}=4x^4+3x^3-x^2-5x-6;\;\; g_{\left(x\right)}=x+3\\[0.2cm]
  4. f(x)=2x6+3x3−4x2−1;    g(x)=x+1\hspace{0.1cm}f_{\left(x\right)}=2x^6+3x^3-4x^2-1;\;\; g_{\left(x\right)}=x+1\\[0.2cm]
  5. f(x)=x3−(2+3)x2+33x−3;    g(x)=x−3\hspace{0.1cm}f_{\left(x\right)}=x^3-\left(2+\sqrt{3}\right)x^2+3\sqrt{3x}-3;\;\; g_{\left(x\right)}=x-\sqrt{3}\\[0.2cm]