Capítulo 3. Ejercicios

En los ejercicios 1 al 5,realice las operaciones entre expresiones algebraicas no racionales indicados.

Para:

$P(x)=x^3-6x^2+2x+4\\[0.3cm] Q(x)=6x^2+2x+4\\[0.3cm] R(x)=-2x^2+4x-5$.

Hallar:

  1. $\hspace{0.1cm}P(x)+Q(x)+R(x)\\[0.3cm]$
  2. $\hspace{0.1cm}P(x)+Q(x)\\[0.3cm]$
  3. $\hspace{0.1cm}P(x)-Q(x)-R(x)\\[0.3cm]$
  4. $\hspace{0.1cm}P(x)-Q(x)\\[0.3cm]$
  5. $\hspace{0.1cm}P(x)*Q(x)\\[0.3cm]$

En los ejercicios del 6 al 15, realice los productos notables indicados.

  1. $\hspace{0.1cm}(a^2 b^2-1)(a^2 b^2+7)$
  2. $\hspace{0.1cm}\left(x^3 y^3-6\right)\left(x^3 y^3+8\right)$
  3. $\hspace{0.1cm}\left(a^x+b^{x+1}\right)^2$
  4. $\hspace{0.1cm}\left(x^{a+1}-3x^{a-2}\right)^2 $
  5. $\hspace{0.1cm}\left(3x^a-5y^m \right)\left(5y^m+3x^a\right)$
  1. $\hspace{0.1cm}\left(a^{x+1}-2b^{x-1}\right)\left(2b^{x-1}+a^{x+1}\right)$
  2. $\hspace{0.1cm}(a^2-2b)^3$
  3. $\hspace{0.1cm}(2x+3y)^3$
  4. $\hspace{0.1cm}\left(\cfrac{3}{4} a^2- \cfrac{1}{2} a+ \cfrac{4}{5}\right)^2$
  5. $\hspace{0.1cm}\left(3-6a+a^2- a^3 \right)^2$

En los ejercicios del 16 al 35, realice las facatorizaciones indicadas.

  1. $\hspace{0.1cm} a\left(x^2+2x+1\right)+\left(x+1\right)$
  2. $\hspace{0.1cm}a^3+b^3+a^2-b^2$
  3. $\hspace{0.1cm}y^4-16$
  4. $\hspace{0.1cm}64a^6n-b^6n$
  5. $\hspace{0.1cm}3^{2n}-3^n-20$
  6. $\hspace{0.1cm} a^2+1-b\left(a^2+1\right)$
  7. $\hspace{0.1cm}t^6+1+t^2+t^4$
  8. $\hspace{0.1cm}2x^2-xy^n-y^{2n}$
  9. $\hspace{0.1cm}x^2-8xy+16y^2-36a^2+12ab-b^2$
  10. $\hspace{0.1cm}8x^6+7x^3-1$
  1. $\hspace{0.1cm}25-x^2-16y^2+8xy$
  2. $\hspace{0.1cm}49x^4-25x^2-9y^2+30xy$
  3. $\hspace{0.1cm}a+4b^2+4ab-x^2-2ax-a^2$
  4. $\hspace{0.1cm}16x^4+8x^2+81$
  5. $\hspace{0.1cm}x^4-93x^2+36$
  6. $\hspace{0.1cm}4a^8-53a^4 b^4+49b^8$
  7. $\hspace{0.1cm}x^8-1$
  8. $\hspace{0.1cm}64x^6-y^6$
  9. $\hspace{0.1cm}64 +a^6$
  10. $\hspace{0.1cm} 8a^3 +27b^6$

En los ejercicios del 36 al 40, realiza las divisiones el método adecuado, determina el cociente y el residuo.

  1. $\hspace{0.1cm}\left(3a^{x+5}+19a^{x+3}-10a^{x+4}-8a^{x+2}+5a^{x+1}\right)\div\left(a^2-3a+5\right)$
  2. $\hspace{0.1cm}\left(8x^4+6x^2-3x+1\right)\div\left(2x^2- x+2\right)$
  3. $\hspace{0.1cm}\left(4x^3+6x^2+5x+6\right)\div\left(2x^2+x+3\right)$
  1. $\hspace{0.1cm}\left(x^3+3x^2-6x+4\right)\div\left( -x^2-2x+7\right)$
  2. $\hspace{0.1cm}\left(2x^3-2x^2+4x+6\right)\div\left( -x^2-5x+5\right)$

En los ejercicios del 41 al 45, racionaliza los denominadodres.

  1. $\hspace{0.1cm}\cfrac{\left(3x-4y-\sqrt{xy}\right)}{\left(3\sqrt{x}-4\sqrt{y}\right)}$
  2. $\hspace{0.1cm}\cfrac{1}{\sqrt[3]{x}-\sqrt[3]{y}}$
  3. $\hspace{0.1cm}\cfrac{a^4-2}{\sqrt{a^4-1}-1}$
  1. $\hspace{0.1cm}\cfrac{2a}{a\sqrt{x}+3\sqrt{a}}$
  2. $\hspace{0.1cm}\cfrac{x^2-2}{\sqrt{x^2-1}-1}$