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Precálculo Sec. 1.5

Precálculo

Ecuaciones polinómicas

En los problemas del 1 al 14, resolver las ecuaciones dadas.

  1. \[ 5(x - 3) = 3(x + 7) + x \]
  2. \[ y - (6 - 2y) = 8(y - 2) \]
  3. \[ \frac{1}{2} (2x-1) = 3\left(x+\frac{1}{4} \right) \]
  4. \[ \frac{x}{4} - \frac{x}{3} = \frac{7}{6} - \frac{4x}{3} \]
  5. \[ \frac{2x-1}{5} = \frac{2+x}{3} \]
  6. \[ \frac{7z+1}{6} + \frac{3}{2} = \frac{3z}{4} \]
  7. \[ \frac{x-1}{3} - \frac{2-3x}{14} = \frac{4x-3}{7} \]
  8. \[ \frac{x-3}{6} - \frac{2x-1}{5} =-1 \]
  9. \[ \frac{x+1}{5} + \frac{x+2}{6} = \frac{x-1}{4} + \frac{x+7}{10} \]
    \[ \begin{aligned} &\frac{x+1}{5} + \frac{x+2}{6} \\[1em] &\hspace{3em}= \frac{x-1}{4} + \frac{x+7}{10} \end{aligned} \]
  10. \[ \frac{5x-2}{3} - \frac{1}{2} (3x-1) = \frac{9x+7}{6} - \frac{2}{9} (5x-1) \]
    \[ \begin{aligned} &\frac{5x-2}{3} - \frac{1}{2} (3x-1) \\[1em] &\hspace{3em}= \frac{9x+7}{6} - \frac{2}{9} (5x-1) \end{aligned} \]
  11. \[ (x-3)^2 = (x-1)^2 \]
  12. \[ (x-5)(x+1) = (x+2)(x-3) + 13 \]
    \[ \begin{aligned} &(x-5)(x+1) \\[1em] &\hspace{4em} = (x+2)(x-3) + 13 \end{aligned} \]
  13. \[ (2x-5) (x-1) + x^2 = (3x-1) (x+2) +1 \]
    \[ \begin{aligned} &(2x-5) (x-1) + x^2 \\[1em] &\hspace{3em} = (3x-1) (x+2) +1 \end{aligned} \]
  14. \[ 8x(x+2) (x-1) = (2x+1)^3 - (2x+3)^2 \]
    \[ \begin{aligned} &8x(x+2) (x-1) \\[1em] &\hspace{3em} = (2x+1)^3 - (2x+3)^2 \end{aligned} \]

En los problemas del 15 al 22, resolver las ecuaciones literales (despejar \(\boldsymbol{x}\)).

  1. \[ 5(5x-a) = a^2 (x-1) \]
  2. \[ a(x+b) + x(b-a) = 2b (2a-x) \]
  3. \[ x^2 + b^2 + b(b-1) = (x+b)^2 \]
  4. \[ (x+a)^3 -2x^3 = 12a^3 - (x-a) \]
  5. \[ \frac{x-a}{b} + \frac{x-b}{a} = 2 \]
  6. \[ \frac{x-3m}{m^2} + \frac{x-2m}{mn} = -\frac{1}{m} \]
  7. \[ \frac{a-x}{a} - \frac{b-x}{b} = \frac{2(a-b)}{ab} \]
  8. \[ \frac{x-a}{a+b} + \frac{a+b}{a-b} = \frac{x+b}{a+b} + \frac{x-b}{a-b} \]

En los problemas del 23 al 26 despejar la variable indicada en términos de las otras.

  1. \[ A = \pi (r^2+rs) ,\; s \]
  2. \[ S = a\frac{1-r^n}{1-r} ,\; a \]
  3. \[ S = \frac{f}{H-h} ,\; h \]
  4. \[ \frac{1}{x} + \frac{1}{y} = \frac{1}{a} ,\; x \]

En los problemas del 27 al 40, resolver las ecuaciones dadas factorizando.

  1. \[ x^2-4x-12=0 \]
  2. \[ x^2-6x+9=0 \]
  3. \[ x^2+24=-11x \]
  4. \[ 2x^2-3x+1=0 \]
  5. \[ 9x^2-17x-2=0 \]
  6. \[ (2x-1)^2 - (x+5)^2 =-19 \]
  7. \[ (x-5)^2 - (x-4)^2 = (2x+3)^2 +12 \]
    \[ \begin{aligned} &(x-5)^2 - (x-4)^2 \\[1em] &\hspace{5em}= (2x+3)^2 +12 \end{aligned} \]
  8. \[ (x-2)^3 - (x+1)^3 = -x(3x+4) -24 \]
    \[ \begin{aligned} &(x-2)^3 - (x+1)^3 \\[1em] &\hspace{4em}= -x(3x+4) -24 \end{aligned} \]
  9. \[ 6x^2 -\frac{5x}{2} = -\frac{1}{4} \]
  10. \[ \frac{2(x+5)}{5} + \frac{x-4}{4} = \frac{x^2-53}{5} \]
  11. \[ x^4-17x^2+16=0 \]
  12. \[ 6y^4 = \frac{y^2}{2} + \frac{1}{4} \]
  13. \[ x^{2/3}+x^{1/3}-6=0 \]
  14. \[ 2x^{2/3}+3x^{1/3}-2=0 \]

En los problemas del 41 al 46, resolver las ecuaciones dadas, mediante la fórmula cuadrática.

  1. \[ 9(x-1)^2=5 \]
  2. \[ \sqrt{3}x-3=4x^2 \]
  3. \[ 2x(2x-3)=-1 \]
  4. \[ (x+15)^2=6x(x+5) \]
  5. \[ x^2-2x-(a^2+2a)=0 \]
  6. \[ \frac{x^2}{2a} - \frac{a+2}{2a} x+1=0 \]

En los problemas del 47 al 60, resolver las ecuaciones fraccionarias dadas.

  1. \[ \frac{x-6}{x} = \frac{x+6}{x-6} + \frac{6}{x} \]
  2. \[ \frac{x}{x+2} - \frac{x}{x-2} = \frac{x-15}{x^2-4} \]
  3. \[ \frac{1}{3x-3} + \frac{1}{4x+4} = \frac{1}{12x-12} \]
  4. \[ \frac{4x+1}{4x-1} = \frac{4x-1}{4x+1} + \frac{6}{16x^2-1} \]
  5. \[ \frac{1}{x} + \frac{1}{4-x} = 1 \]
  6. \[ \frac{x}{1+x} + \frac{1}{1-x} = 0 \]
  7. \[ \frac{3y-2}{3y+2} = \frac{2y+3}{4y-1} \]
  8. \[ \frac{ x+5 }{ (x-1)(x+2) } = \frac{2x}{x+2} \]
  9. \[ \frac{1}{x-1} - \frac{1}{x-2} = \frac{1}{x-3} \]
  10. \[ \frac{3x}{x-2} - \frac{1}{x^2-4} = 2 \]
  11. \[ \frac{1}{x^2} + \frac{2}{x} -15 = 0 \]
  12. \[ \frac{12}{x-1} + \frac{12}{x} = 10 \]
  13. \[ \frac{2x}{x-1} = \frac{8}{x-1} - \frac{5}{x} \]
  14. \[ \frac{1}{x^2-4} + \frac{2x+3}{x+2} + \frac{x+3}{x-2} = 0 \]

En los problemas del 61 al 76, resolver las ecuaciones radicales dadas. Eliminar las soluciones extrañas.

  1. \[ 5-\sqrt{2x+3}= \]
  2. \[ \sqrt{ \frac{x}{18} + 1 } = \frac{2}{3} \]
  3. \[ (5x-1)^{1/2}=7 \]
  4. \[ (y+9)^{3/2} = 4^3 \]
  5. \[ \sqrt{x^2-5} = 5-x \]
  6. \[ \sqrt{z+7} - \sqrt{z} = 1 \]
  7. \[ \sqrt{ 9x^2 - 10x } = 3x-2 \]
  8. \[ \sqrt{ \frac{1}{x} } - \sqrt{ \frac{8}{4x+1} } = 0 \]
  9. \[ \sqrt{4x+1} + 1 = 2x \]
  10. \[ \sqrt{x^2+5} = 2x-1 \]
  11. \[ \sqrt{x+5} = 2\sqrt{x} - 1 \]
  12. \[ \sqrt{x} + \sqrt{x-3} = \sqrt{x+5} \]
  13. \[ \sqrt{ x + \sqrt{x+8} } = 2\sqrt{x} \]
  14. \[ \sqrt{ x + \sqrt{x+8} } = 2\sqrt{x} \]
  15. \[ \sqrt{ x + \sqrt{x+8} } = 2\sqrt{x} \]
  16. \[ \frac{x}{2} = \frac{ \sqrt{x+2} - \sqrt{x-2} }{ \sqrt{x+2} + \sqrt{x-2} } \]

En los problemas 77 y 78, resolver efectuando un cambio de variable.

  1. \[ \left( \frac{3x}{x+1} \right)^2 - \frac{6x}{x+1} = 8 \]
  2. \[ \sqrt[3]{ \frac{5x+4}{x-1} } + \sqrt[3]{ \frac{x-1}{5x+4} } = \frac{5}{2} \]

En los problemas 79 y 80, usando el teorema del residuo, hallar el residuo cuando se divide:

  1. \[ 3x^4 - 5x^3 - 4x^2 + 3x - 2, \; \text{ entre } (x-2) \]
    \[ \begin{aligned} &3x^4 - 5x^3 - 4x^2 + 3x - 2, \\[1em] &\hspace{4em}\text{entre } (x-2) \end{aligned} \]
  2. \[ x^3 - 6x^2 +11x - 6, \; \text{ entre } (x+2) \]
    \[ \begin{aligned} &x^3 - 6x^2 +11x - 6, \\[1em] &\hspace{4em} \text{entre } (x+2) \end{aligned} \]

En los problemas del 81 al 88, hallar las raíces de la ecuación dada y factorice el polinomio correspondiente.

  1. \[ x^3 + 2x^2 - x - 2 = 0 \]
  2. \[ x^3 - 3x^2 + 2 = 0 \]
  3. \[ 4x^3 - 7x^2 + 3 = 0 \]
  4. \[ 2x^3 - 2x^2 - 11x + 2 = 0 \]
  5. \[ x^4 - x^3 - 5x^2 + 3x + 6 = 0 \]
  6. \[ 3x^4 + 5x^3 - 5x^2 - 5x + 2 = 0 \]
  7. \[ x^5 -3x^4 -5x^3 +15x^2+4x -12 = 0 \]
    \[ \begin{aligned} x^5 -3x^4 -5x^3 &+15x^2 \\[1em] &+ 4x -12 = 0 \end{aligned} \]
  8. \[ x^5 + 4x^4 - 4x^3 -34x^2 -45x-18 = 0 \]
    \[ \begin{aligned} x^5 + 4x^4 - 4x^3 &-34x^2 \\[1em] &-45x-18 = 0 \end{aligned} \]

En los problemas del 89 al 91, usar el teorema del factor para probar que:

  1. \(x - a\) es un factor de \(x^n - a^n\), para todo entero positivo \(n\).
  2. \(x + a\) es un factor de \(x^n - a^n\), para todo entero positivo par \(n\).
  3. \(x + a\) es un factor de \(x^n + a^n\) , para todo entero positivo impar \(n\).
  1. \[ x=36 \]
  2. \[ y = 2 \]
  3. \[ x = -\frac{5}{8} \]
  4. \[ x = 2 \]
  5. \[ x = 13 \]
  6. \[ z = -4 \]
  7. \[ x = -2 \]
  8. \[ x = 3 \]
  9. \[ x = -5 \]
  10. \[ x = -7 \]
  11. \[ x = 2 \]
  12. \[ x = -4 \]
  13. \[ x = \frac{1}{2} \]
  14. \[ x = \frac{4}{5} \]
  15. \[ x = \frac{a}{a + 5} \]
  16. \[ x = a \]
  17. \[ x = \frac{b – 1}{2} \]
  18. \[ x = 2a \]
  19. \[ x = a + b \]
  20. \[ x = 2m \]
  21. \[ x = 2 \]
  22. \[ x = 3b \]
  23. \[ s = \frac{A + \pi r^2}{\pi r} \]
  24. \[ a = \frac{ S (I – r) }{1 – r^n} \]
  25. \[ h = \frac{HS – f}{S} \]
  26. \[ x = \frac{ay}{y – a} \]
  27. \[ -2, \; 6 \]
  28. \[ 3 \]
  29. \[ -3,\; -8 \]
  30. \[ 1, \; \frac{1}{2} \]
  31. \[ 2, \; -\frac{1}{9} \]
  32. \[ 5, \; -\frac{1}{3} \]
  33. \[ -2, \; -\frac{3}{2} \]
  34. \[ 3, \; -\frac{5}{6} \]
  35. \[ \frac{1}{4}, \; \frac{1}{6} \]
  36. \[ 8, -\frac{19}{4} \]
  37. \[ -1, \, 1, \, -4, \, 4 \]
  38. \[ -\frac{1}{2}, \; \frac{1}{2} \]
  39. \[ -27, \; 8 \]
  40. \[ \frac{1}{8}, \; -8 \]
  41. \[ 1 – \frac{ 1 }{3} \sqrt{5}, \, 1 + \frac{1}{3} \sqrt{5} \]
  42. \[ \frac{1}{2} \sqrt{3} \]
  43. \[ \frac{3}{4}- \frac{\sqrt{5}}{4}, \, \frac{3}{4} + \frac{\sqrt{5}}{4} \]
  44. \[ -3 \sqrt{5}, \, 3 \sqrt{5} \]
  45. \[ -a , \; a + 2 \]
  46. \[ a, \; 2 \]
  47. \[ x = 3 \]
  48. \[ x = 0 \]
  49. \[ x = 0 \]
  50. \[ x = \frac{3}{8} \]
  51. \[ 2 \]
  52. \[ 1 – \sqrt{2}, \, 1 + \sqrt{2} \]
  53. \[ 2 – \frac{1}{3} \sqrt{42}, \, 2 + \frac{1}{3}\sqrt{42} \]
  54. \[ \frac{5}{2} \]
  55. \[ 1 – \sqrt{2}, \, 1 + \sqrt{2} \]
  56. \[ – 3 – \sqrt{2}, \, – 3 + \sqrt{2} \]
  57. \[ \frac{1}{3}, \, -\frac{1}{5} \]
  58. \[ 3 , \, \frac{2}{5} \]
  59. \[ -1, \, \frac{5}{2} \]
  60. \[ -1 \]
  61. \[ 11 \]
  62. \[ -10 \]
  63. \[ 10 \]
  64. \[ y = 7 \]
  65. \[ x = 3 \]
  66. \[ z = 9 \]
  67. \[ x = 2 \]
  68. \[ \frac{1}{4} \]
  69. \[ 2 \]
  70. \[ 2 \]
  71. \[ 4 \]
  72. \[ 4 \]
  73. \[ 1 \]
  74. \[ 2 \]
  75. \[ \frac{9}{16} \]
  76. \[ 2 \]
  77. \[ -4, \; -\frac{2}{5} \]
  78. \[ 4, \, -\frac{11}{13} \]
  79. \[ -4 \]
  80. \[ -60 \]

  81. raíces: \(1, \, -2, \, -1;\)   \((x – 1)(x + 2)(x + 1)\)

  82. raíces: \(1, \, 1 + \sqrt{3}, \, 1 – \sqrt{3}\);   \((x – 1) \left( x – 1 – \sqrt{3} \right) \left( x – 1 + \sqrt{3} \right)\)

  83. raíces: \(1, \, -\frac{3}{2}, \, \frac{1}{2}\);   \(4(x – 1) \left( x + \frac{3}{2} \right) \left( x – \frac{1}{2} \right)\)

  84. raíces: \(-2, \, \frac{3}{2} + \frac{\sqrt{7}}{2}\), \(\frac{3}{2} – \frac{\sqrt{7}}{2}\);   \(2 (x + 2) \left( x – \frac{3}{2} + \frac{\sqrt{7}}{2} \right) \left( x – \frac{3}{2} – \frac{\sqrt{7}}{2} \right)\)

    raíces: \(-2, \, \frac{3}{2} + \frac{\sqrt{7}}{2}\), \(\frac{3}{2} – \frac{\sqrt{7}}{2}\);

    \[ \begin{aligned} &2 (x + 2) \left( x – \frac{3}{2} + \frac{\sqrt{7}}{2} \right) \\[.5em] &\hspace{6em} \times \left( x – \frac{3}{2} – \frac{\sqrt{7}}{2} \right) \end{aligned} \]

  85. raíces: \(-1, \, 2, \, \sqrt{3}, \, -\sqrt{3}\);   \((x + 1)(x – 2) \left( x – \sqrt{3} \right) \left( x + \sqrt{3} \right)\)

  86. raíces: \(1, \, -1, \, -2, \, \frac{1}{3}\);   \(3(x – 1)(x + 1)(x + 2) \left( x – \frac{1}{3} \right)\)

  87. raíces: \(-1, \, -2, \, 1, \, 2, \, 3\);   \((x + 1)(x+2)(x-1)(x-2)(x-3)\)

  88. raíces: \(-1, \, -2, \, -3, \, 3\);   \((x + 1)^2 (x + 2) (x + 3)(x – 3)\)

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