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Precálculo Sec. 1.4

Precálculo

Algo de algebra

En los problemas del 1 al 16, efectuar los productos indicados, usando las formulas de productos notables.

  1. \[ \left( 2x+\sqrt{5} \right)\left( 2x-\sqrt{5} \right) \]
  2. \[ \left( 2\sqrt{x}+\sqrt{y} \right)\left( 2\sqrt{x}-\sqrt{y} \right) \]
  3. \[ \left( 3x^2+4y^3\right)\left( 3x^2-4y^3 \right) \]
  4. \[ \left( \sqrt{h+1}+1 \right) \left( \sqrt{h+1}-1 \right) \]
  5. \[ \left( \sqrt{x}+ \frac{1}{y} \right) \left( \sqrt{x} - \frac{1}{y} \right) \]
  6. \[ (a+b+c)(a+b-c) \]
  7. \[ (4x+5)^2 \]
  8. \[ (2x-5y)^2 \]
  9. \[ \left( x-x^{-1} \right)^2 \]
  10. \[ \left( x^3-x^{-3} \right)^2 \]
  11. \[ (4x+y)^3 \]
  12. \[ \left( a^2 + b^2\right)^3 \]
  13. \[ \left( x^2-y \right)^3 \]
  14. \[ \left( \sqrt[3]{x}+\sqrt[3]{y} \right)^3 \]
  15. \[ (x-5)^2(x+5)^2 \]
  16. \[ (2x-y)(2x+y)\left( 4x^2+y^2 \right) \]

En los problemas del 17 al 56, factorizar las expresiones dadas.

  1. \[ 7x^3-63x^2 \]
  2. \[ 8x^2y^2z^3-24xy^3z^2-4x^3y^4z^3 \]
  3. \[ x^3-2x^2-4x+8 \]
  4. \[ 4y^2+16y+12xy+48x \]
  5. \[ x^2y^2-y^2-4x+4 \]
  6. \[ 2a^2x-5a^2y+15by-6bx \]
  7. \[ x^2+2x-48 \]
  8. \[ x^2-4x-5 \]
  9. \[ y^2+28y-29 \]
  10. \[ x^2+15x-216 \]
  11. \[ x^4-2x^2-80 \]
  12. \[ a^2b^2+ab-12 \]
  13. \[ 3x^2 + 7x + 4 \]
  14. \[ 5y^2 + 10y - 75 \]
  15. \[ 5a^2x^2 + 4ax - 12 \]
  16. \[ 9x^2 - 15x - 50 \]
  17. \[ 4x^2y^2 + 11xy^2 + 6y^2 \]
  18. \[ 25x^4 - 10x^2 + 1 \]
  19. \[ 25x^2 - 36y^4 \]
  20. \[ 63x^4 - 7x^2 \]
  21. \[ 45x^2y^2 - 5x^4 \]
  22. \[ \frac{x^2}{36} - \frac{y^2}{25} \]
  23. \[ 16x^{2n} - \frac{1}{49} \]
  24. \[ (a - b)^2 - 9 \]
  25. \[ (a + b)^2 - (a - b)^2 \]
  26. \[ (x - 1)^2 - (y - 2)^2 \]
  27. \[ x^2 - y^2 - 6y - 9 \]
  28. \[ 9(a - b)^2 - 4(a + b)^2 \]
  29. \[ a^4 - 2a^2 + 1 \]
  30. \[ 16x^2 - 24xy + 9y^2 \]
  31. \[ 400x^4+ 40x^2 + 1 \]
  32. \[ \frac{x^2}{9} + \frac{2x}{3} + 1 \]
  33. \[ \frac{4x^2}{25} - \frac{x}{5} + \frac{1}{16} \]
  34. \[ 8x^3- y^3 \]
  35. \[ 27a^3 + 64b^3 \]
  36. \[ 5x^3y^3 + 5 \]
  37. \[ x^5- 125x^2 \]
  38. \[ (x + y)^3 - 1 \]
  39. \[ (x - y)^3 - 8 \]
  40. \[ (x + 1)^3 - (x - 2)^3 \]

En los problemas del 57 al 68, simplificar las fracciones dadas.

  1. \[ \frac{60a^3b^2-45a^2b}{15a^2b} \]
  2. \[ \frac{x^2-3x}{3-x} \]
  3. \[ \frac{a^2-1}{a+1} \]
  4. \[ \frac{x^2-x-20}{x^2+2x-8} \]
  5. \[ \frac{2x^2+x-6}{2x-3} \]
  6. \[ \frac{x^2+x-2}{2x^2+6x+4} \]
  7. \[ \frac{x^2-y^2}{x^2+2xy+y^2} \]
  8. \[ \frac{x^2-4xy+4y^2}{x^3-8y^3} \]
  9. \[ \frac{(3-a)^2}{27-a^3} \]
  10. \[ \frac{x^3+1}{x^4-x^3+x-1} \]
  11. \[ \frac{y+8y^2+16y^3}{6y^2+25y^3+4y^4} \]
  12. \[ \frac{x^2-y^2}{x^2-6y-xy+6x} \]

En los problemas del 69 al 80, racionalizar el denominador.

  1. \[ \frac{2}{1-\sqrt{2}} \]
  2. \[ \frac{h}{\sqrt{3+h}-\sqrt{3}} \]
  3. \[ \frac{2a}{\sqrt{a+1}-\sqrt{a-1}} \]
  4. \[ \frac{3\sqrt{2}}{7\sqrt{2}-6\sqrt{3}} \]
  5. \[ \frac{\sqrt{x}+\sqrt{a}}{\sqrt{x}+2\sqrt{a}} \]
  6. \[ \frac{5}{\sqrt{x-3}-\sqrt{x-13}} \]
  7. \[ \frac{3}{\sqrt[3]{7}+\sqrt[3]{2}} \]
  8. \[ \frac{16x-2}{2\sqrt[3]{x}-1} \]
  9. \[ \frac{70x-16}{2\sqrt[3]{x-1}+3\sqrt[3]{x}} \]
  10. \[ \frac{3x-9y}{\sqrt[3]{x^2}+\sqrt[3]{3xy}+\sqrt[3]{9y^2}} \]
  11. \[ \frac{8-x}{\sqrt{2-\sqrt[3]{x}}} \]
  12. \[ \frac{2x-1}{\sqrt{2\sqrt{x}+\sqrt{2}}} \]

En los problemas del 81 al 83, racionalizar el numerador.

  1. \[ \frac{3+\sqrt{5}}{4} \]
  2. \[ \frac{\sqrt{a+2}-\sqrt{a}}{2} \]
  3. \[ \frac{\sqrt{a-1+h}-\sqrt{a-1}}{h} \]

En los problemas del 84 al 104, efectuar las operaciones indicadas.

  1. \[ \frac{3a}{a+1} + \frac{2a}{a-1} \]
  2. \[ \frac{x+y}{x-y} - \frac{x-y}{x+y} \]
  3. \[ \frac{12}{x^2-9} - \frac{2}{x-3} + 1 \]
  4. \[ \frac{x-2}{x^2-x-2} - \frac{2}{x^2-1} \]
  5. \[ \frac{1}{x+1} + \frac{2}{x-1} - \frac{1}{x^2-1} \]
  6. \[ \frac{x+5}{x^2+2x+1} + \frac{x}{x^2-4x-5} + \frac{1}{x-5} \]
    \[ \begin{aligned} \frac{x+5}{x^2+2x+1} &+ \frac{x}{x^2-4x-5} \\[1em] &+ \frac{1}{x-5} \end{aligned} \]
  7. \[ \frac{x}{x^2-x-2} - \frac{6}{x^2+5x-14} - \frac{1}{x^2+8x+7} \]
    \[ \begin{aligned} \frac{x}{x^2-x-2} &- \frac{6}{x^2+5x-14} \\[1em] &- \frac{1}{x^2+8x+7} \end{aligned} \]
  8. \[ \frac{x^2}{y^2-x^2} \times \frac{xy-x^2}{xy} \]
  9. \[ \frac{x^2+4x}{3x-2} \times \frac{9x^2-4}{x^2-16} \]
  10. \[ \frac{x^3-8}{a^3-1} \times \frac{a^2+a+1}{x^2+2x+4} \]
  11. \[ \begin{aligned} \frac{ x^2 + xy - 2y^2 }{ x^2 - 2xy - 8 y^2 } &\times \frac{ x^2 + 2 xy }{ x^2 + 4 xy} \\[1em] &\times \frac{ x^2 - 16 y^2 }{ x + 2y} \end{aligned} \]
    \[ \frac{ x^2 + xy - 2y^2 }{ x^2 - 2xy - 8 y^2 } \times \frac{ x^2 + 2 xy }{ x^2 + 4 xy} \times \frac{ x^2 - 16 y^2 }{ x + 2y} \]
  12. \[ \frac{a^2-ab-6b^2}{b^2+ab} \div \frac{a^2-4b^2}{a^2+ab} \]
  13. \[ \frac{x^4-x}{x^2+6x+8} \div \frac{2x^2-x-1}{2x^2+9x+4} \]
  14. \[ \frac{25x^3-x}{25x^2-10x+1} \div \frac{6x^2+13x+6}{15x^2+7x-2} \]
    \[ \begin{aligned} &\frac{25x^3-x}{25x^2-10x+1} \\[1em] &\hspace{3em} \div \frac{6x^2+13x+6}{15x^2+7x-2} \end{aligned} \]
  15. \[ \left( \frac{x+1}{3x-3} \times \frac{6x-6}{2x+4} \right) \div \frac{x^2+x}{x^2+x-2} \]
    \[ \begin{aligned} &\left( \frac{x+1}{3x-3} \times \frac{6x-6}{2x+4} \right) \\[1em] &\hspace{3em} \div \frac{x^2+x}{x^2+x-2} \end{aligned} \]
  16. \[ \frac{3x^2+3}{2x-4} \div \left( \frac{3x+6}{2x-6} \times \frac{x^3+x}{3x-6} \right) \]
    \[ \begin{aligned} &\frac{3x^2+3}{2x-4} \\[1em] &\hspace{2em} \div \left( \frac{3x+6}{2x-6} \times \frac{x^3+x}{3x-6} \right) \end{aligned} \]
  17. \[ \left( 1-\frac{a^3}{b^3} \right) \left( b+\frac{ab}{b-a} \right) \]
  18. \[ \left( x+\frac{4x^2+20x}{x^2-25} \right) \left( x+2-\frac{28}{x-1} \right) \]
    \[ \begin{aligned} &\left( x+\frac{4x^2+20x}{x^2-25} \right) \\[1em] &\hspace{2em} \times \left( x+2-\frac{28}{x-1} \right) \end{aligned} \]
  19. \[ \left( \frac{x^2}{x^2-y^2}-1 \right) \left( \frac{x}{y}-1 \right) \left( \frac{y}{x}+1 \right) \]
    \[ \begin{aligned} &\left( \frac{x^2}{x^2-y^2}-1 \right) \left( \frac{x}{y}-1 \right) \\[1em] &\hspace{2em} \times \left( \frac{y}{x}+1 \right) \end{aligned} \]
  20. \[ \left( \frac{x^2}{x+1} -x+1 \right) \div \left( \frac{2}{x^2-1} +1 \right) \]
    \[ \begin{aligned} &\left( \frac{x^2}{x+1} -x+1 \right) \\[1em] &\hspace{2em}\div \left( \frac{2}{x^2-1} +1 \right) \end{aligned} \]
  21. \[ \left( \frac{2a+1}{a^2+2} -a \right) \div \left( \frac{a+1}{a} -a^2-1 \right) \]
    \[ \begin{aligned} &\left( \frac{2a+1}{a^2+2} -a \right) \\[1em] &\hspace{2em} \div \left( \frac{a+1}{a} -a^2-1 \right) \end{aligned} \]

En los problemas del 105 al 109, simplificar las fracciones compuestas dadas.

  1. \[ \cfrac{ \cfrac{1}{x} -x^2 }{ \cfrac{1}{x} -1 } \]
  2. \[ \cfrac{ \cfrac{a}{b^2} - \cfrac{b}{a^2} } { \cfrac{1}{b^2} - \cfrac{1}{a^2} } \]
  3. \[ a- \cfrac{b} { \cfrac{a}{b} + \cfrac{b}{a} } \]
  4. \[ 1- \cfrac{1}{ 1- \cfrac{1} { 1- \cfrac{1}{x^2} } } \]
  5. \[ \cfrac{ 1- \cfrac{1}{a-2} }{ a+3 - \cfrac{24}{a+1} } \]

Respuestas

  1. \[ 4x^2 – 5 \]
  2. \[ 4x – y \]
  3. \[ 9x^4 – 16 y^6 \]
  4. \[ h \]
  5. \[ x – \frac{1}{y^2} \]
  6. \[ a^2 + b^2 – c^2 + 2ab \]
  7. \[ 16x^2 + 40x + 25 \]
  8. \[ 4x^2 -20xy +25y^2 \]
  9. \[ x^2 – 2 + \frac{1}{x^2} \]
  10. \[ x^6 – 2 + \frac{1}{x^6} \]
  11. \[ 64x^3+ 48x^2 y + 12xy^2 + y^3 \]
  12. \[ a^6 + 3 a^4 b^2 + 3 a^2 b^4 + b^6 \]
  13. \[ x^6 – 3x^4 y + 3x^2y^2 – y^3 \]
  14. \[ x + 3 \sqrt[3]{x^2 y}+ 3 \sqrt[3]{x y^2} + y \]
  15. \[ x^4 – 50x^2 + 625 \]
  16. \[ 16x^4 – y^4 \]
  17. \[ 7 x^2 (x – 9) \]
  18. \[ 4xy^2 z^2 \left( 2xz – 6y – x^2 y^2 z \right) \]
  19. \[ (x – 2)^2 (x + 2) \]
  20. \[ 4(y + 4)(y + 3x) \]
  21. \[ (x – 1) \left( x y^2 + y^2 – 4 \right) \]
  22. \[ (2x – 5y) \left( a^2 – 3b \right) \]
  23. \[ (x + 8)(x – 4) \]
  24. \[ (x – 5)(x + 1) \]
  25. \[ (xy + 29)(y – 1) \]
  26. \[ (x + 24)(x – 9) \]
  27. \[ \left( x^2 – 10 \right) \left( x^2 + 8 \right) \]
  28. \[ (ab + 4)(ab – 3) \]
  29. \[ (3x + 4)(x + 1) \]
  30. \[ 5(y + 5)(y – 3) \]
  31. \[ (ax + 2)(5ax – 6) \]
  32. \[ (3x – 10)(3x + 5) \]
  33. \[ y^2 (x + 2)(4x + 3) \]
  34. \[ \left( 5x^2 – 1 \right)^2 \]
  35. \[ \left( 5x + 6y^2 \right) \left( 5x – 6y^2 \right) \]
  36. \[ 7x^2(3x + 1)(3x – 1) \]
  37. \[ 5x^2 (3y + x)(3y – x) \]
  38. \[ \left( \frac{x}{6} + \frac{y}{5} \right) \left( \frac{x}{6} – \frac{y}{5} \right) \]
  39. \[ \left( 4x^n + \frac{1}{7} \right) \left( 4x^n – \frac{1}{7} \right) \]
  40. \[ (a – b + 3)(a – b – 3) \]
  41. \[ 4ab \]
  42. \[ (x + y – 3)(x – y + 1) \]
  43. \[ (x + y + 3)(x – y – 3) \]
  44. \[ (5a – b)(a – 5b) \]
  45. \[ \left( a^2 – 1 \right)^2 \]
  46. \[ (4x – 3y)^2 \]
  47. \[ \left( 20x^2 + 1 \right)^2 \]
  48. \[ \left( \frac{x}{3} + 1 \right)^2 \]
  49. \[ \left( \frac{2x}{5} – \frac{1}{4}\right)^2 \]
  50. \[ (2x – y) \left( 4x^2 + 2xy + y^2 \right) \]
  51. \[ (3a + 4b) \left( 9a^2 – 12ab + 16b^2 \right) \]
  52. \[ 5(xy + 1) \left( x^2y^2 – xy + 1 \right) \]
  53. \[ x^2 (x – 5) \left( x^2 + 5x + 25 \right) \]
  54. \[ (x + y – 1)\left( x^2 + 2xy + y^2 + x + y + 1 \right) \]
    \[ \begin{aligned} &(x + y – 1) \\[.5em] &\hspace{1em} \times \left( x^2 + 2xy + y^2 + x + y + 1 \right) \end{aligned} \]
  55. \[ (x – y – 2) \left(x^2 – 2xy + y^2 + 2x – 2y + 4 \right) \]
    \[ \begin{aligned} &(x – y – 2) \\[.5em] &\hspace{.5em} \times \left(x^2 – 2xy + y^2 + 2x – 2y + 4 \right) \end{aligned} \]
  56. \[ 9 \left( x^2 – x + 1 \right) \]
  57. \[ 4ab – 3 \]
  58. \[ -x \]
  59. \[ a – 1 \]
  60. \[ \frac{x – 5}{x – 2} \]
  61. \[ 2x + 4 \]
  62. \[ \frac{x – 1}{2x + 2} \]
  63. \[ \frac{x-y}{x+y} \]
  64. \[ \frac{x – 2y}{ x^2 + 2xy + 4y^2} \]
  65. \[ \frac{3 – a}{ 9 + 3a + a^2} \]
  66. \[ \frac{1}{x-1} \]
  67. \[ \frac{4y +1}{ y^2 + 6y} \]
  68. \[ \frac{x + y}{6 + x} \]
  69. \[ – 2 – 2 \sqrt{2} \]
  70. \[ \sqrt{3 + h} + \sqrt{3} \]
  71. \[ a \sqrt{a + 1} + a \sqrt{a – 1} \]
  72. \[ – \frac{21 + 9\sqrt{6}}{5} \]
  73. \[ \frac{ x – \sqrt{a x} – 2a }{x – 4a} \]
  74. \[ \frac{ \sqrt{x – 3} + \sqrt{x – 13}}{2} \]
  75. \[ \frac{ \sqrt[3]{49}- \sqrt[3]{14} + \sqrt[3]{4} }{3} \]
  76. \[ 8 \sqrt[3]{x^2} + 4 \sqrt[3]{x} + 2 \]
  77. \[ 8 \sqrt[3]{(x – 1)^2} – 12 \sqrt[3]{ x^2 – x } + 18 \sqrt[3]{x^2} \]
    \[ \begin{aligned} 8 \sqrt[3]{(x – 1)^2} &- 12 \sqrt[3]{ x^2 – x } \\[.5em] &+ 18 \sqrt[3]{x^2} \end{aligned} \]
  78. \[ 3 \sqrt[3]{x} – 3 \sqrt[3]{3y} \]
  79. \[ \left( \sqrt{ 2 – \sqrt[3]{x}} \right) \left( 4 + 2 \sqrt[3]{x} + \sqrt[3]{x^2} \right) \]
  80. \[ \sqrt{ 2 \sqrt[3]{x^2} + \sqrt{2} x } – \sqrt{ 4 \sqrt{x} + \sqrt{8}} \]
  81. \[ \frac{1}{3 – \sqrt{5}} \]
  82. \[ \frac{1}{\sqrt{a + 2} + \sqrt{a}} \]
  83. \[ \frac{ 1 }{ \sqrt{ a – 1 + h } + \sqrt{a – 1}} \]
  84. \[ \frac{5 a^2 – a}{a^2 – 1} \]
  85. \[ \frac{ 4xy }{ x^2 – y^2 } \]
  86. \[ \frac{x + 1}{x + 3} \]
  87. \[ \frac{x – 3}{x^2 – 1} \]
  88. \[ \frac{3x}{x^2 – 1} \]
  89. \[ \frac{3x^2 + 3x – 24}{ x^3 – 3x^2 – 9x – 5 } \]
  90. \[ \frac{ x + 2 }{ x^2 + 8x + 7 } \]
  91. \[ \frac{x^2 y – x^3} { y^3 – x^2 y } \]
  92. \[ \frac{ 3 x^2 + 2x }{x – 4} \]
  93. \[ \frac{x – 2}{a – 1} \]
  94. \[ \frac{ x^3 + 8y^3 – 3y x^2 – 6xy^2 }{ x^2 – 2xy – 8 y^2} \]
  95. \[ \frac{ a^3 – a^2 b – 6 a b^2 } { a^2 b – 4 b^3 } \]
  96. \[ \frac{x^3 + x^2 + x}{ x + 2} \]
  97. \[ \frac{5x^2 + x} {2x + 3} \]
  98. \[ \frac{x – 1}{x} \]
  99. \[ \frac{ 3x – 9 }{x^2 + 2x} \]
  100. \[ \frac{ b^2 + ab + a^2}{b} \]
  101. \[ x^2 + 6x \]
  102. \[ \frac{y}{x} \]
  103. \[ \frac{x – 1}{ x^2 + 1} \]
  104. \[ \frac{ a }{ a^2 + 2} \]
  105. \[ x^2 + x + 1 \]
  106. \[ \frac{a^2 + ab + b^2}{a + b} \]
  107. \[ \frac{a^3}{ a^2 + b^2 } \]
  108. \[ x^2 \]
  109. \[ \frac{a+1}{a^2 + 5a – 14} \]
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