Capítulo 3. Ejercicios

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

Para:

P(x)=x3−6x2+2x+4Q(x)=6x2+2x+4R(x)=−2x2+4x−5P(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. P(x)+Q(x)+R(x)\hspace{0.1cm}P(x)+Q(x)+R(x)\\[0.3cm]
  2. P(x)+Q(x)\hspace{0.1cm}P(x)+Q(x)\\[0.3cm]
  3. P(x)−Q(x)−R(x)\hspace{0.1cm}P(x)-Q(x)-R(x)\\[0.3cm]
  4. P(x)−Q(x)\hspace{0.1cm}P(x)-Q(x)\\[0.3cm]
  5. P(x)∗Q(x)\hspace{0.1cm}P(x)*Q(x)\\[0.3cm]

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

  1. (a2b2−1)(a2b2+7)\hspace{0.1cm}(a^2 b^2-1)(a^2 b^2+7)
  2. (x3y3−6)(x3y3+8)\hspace{0.1cm}\left(x^3 y^3-6\right)\left(x^3 y^3+8\right)
  3. (ax+bx+1)2\hspace{0.1cm}\left(a^x+b^{x+1}\right)^2
  4. (xa+1−3xa−2)2\hspace{0.1cm}\left(x^{a+1}-3x^{a-2}\right)^2
  5. (3xa−5ym)(5ym+3xa)\hspace{0.1cm}\left(3x^a-5y^m \right)\left(5y^m+3x^a\right)
  1. (ax+1−2bx−1)(2bx−1+ax+1)\hspace{0.1cm}\left(a^{x+1}-2b^{x-1}\right)\left(2b^{x-1}+a^{x+1}\right)
  2. (a2−2b)3\hspace{0.1cm}(a^2-2b)^3
  3. (2x+3y)3\hspace{0.1cm}(2x+3y)^3
  4. (34a2−12a+45)2\hspace{0.1cm}\left(\cfrac{3}{4} a^2- \cfrac{1}{2} a+ \cfrac{4}{5}\right)^2
  5. (3−6a+a2−a3)2\hspace{0.1cm}\left(3-6a+a^2- a^3 \right)^2

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

  1. a(x2+2x+1)+(x+1)\hspace{0.1cm} a\left(x^2+2x+1\right)+\left(x+1\right)
  2. a3+b3+a2−b2\hspace{0.1cm}a^3+b^3+a^2-b^2
  3. y4−16\hspace{0.1cm}y^4-16
  4. 64a6n−b6n\hspace{0.1cm}64a^6n-b^6n
  5. 32n−3n−20\hspace{0.1cm}3^{2n}-3^n-20
  6. a2+1−b(a2+1)\hspace{0.1cm} a^2+1-b\left(a^2+1\right)
  7. t6+1+t2+t4\hspace{0.1cm}t^6+1+t^2+t^4
  8. 2x2−xyn−y2n\hspace{0.1cm}2x^2-xy^n-y^{2n}
  9. x2−8xy+16y2−36a2+12ab−b2\hspace{0.1cm}x^2-8xy+16y^2-36a^2+12ab-b^2
  10. 8x6+7x3−1\hspace{0.1cm}8x^6+7x^3-1
  1. 25−x2−16y2+8xy\hspace{0.1cm}25-x^2-16y^2+8xy
  2. 49x4−25x2−9y2+30xy\hspace{0.1cm}49x^4-25x^2-9y^2+30xy
  3. a+4b2+4ab−x2−2ax−a2\hspace{0.1cm}a+4b^2+4ab-x^2-2ax-a^2
  4. 16x4+8x2+81\hspace{0.1cm}16x^4+8x^2+81
  5. x4−93x2+36\hspace{0.1cm}x^4-93x^2+36
  6. 4a8−53a4b4+49b8\hspace{0.1cm}4a^8-53a^4 b^4+49b^8
  7. x8−1\hspace{0.1cm}x^8-1
  8. 64x6−y6\hspace{0.1cm}64x^6-y^6
  9. 64+a6\hspace{0.1cm}64 +a^6
  10. 8a3+27b6\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. (3ax+5+19ax+3−10ax+4−8ax+2+5ax+1)÷(a2−3a+5)\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. (8x4+6x2−3x+1)÷(2x2−x+2)\hspace{0.1cm}\left(8x^4+6x^2-3x+1\right)\div\left(2x^2- x+2\right)
  3. (4x3+6x2+5x+6)÷(2x2+x+3)\hspace{0.1cm}\left(4x^3+6x^2+5x+6\right)\div\left(2x^2+x+3\right)
  1. (x3+3x2−6x+4)÷(−x2−2x+7)\hspace{0.1cm}\left(x^3+3x^2-6x+4\right)\div\left( -x^2-2x+7\right)
  2. (2x3−2x2+4x+6)÷(−x2−5x+5)\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. (3x−4y−xy)(3x−4y)\hspace{0.1cm}\cfrac{\left(3x-4y-\sqrt{xy}\right)}{\left(3\sqrt{x}-4\sqrt{y}\right)}
  2. 1x3−y3\hspace{0.1cm}\cfrac{1}{\sqrt[3]{x}-\sqrt[3]{y}}
  3. a4−2a4−1−1\hspace{0.1cm}\cfrac{a^4-2}{\sqrt{a^4-1}-1}
  1. 2aax+3a\hspace{0.1cm}\cfrac{2a}{a\sqrt{x}+3\sqrt{a}}
  2. x2−2x2−1−1\hspace{0.1cm}\cfrac{x^2-2}{\sqrt{x^2-1}-1}