Ejercicios Algebra

Ejercicio 2.3
Encuentre todas las raíces de la ecuación dada.
1) 3x3-20x2+29x+12=0
+ | - | i |
2 | 1 | 0 |
0 | 1 | 2 |
pq=±1, ±2, ±3, ±4, ±6, ±12±1, ±3
pq=±1, ±13,±2, ±23,±3±1, ±4, ±43,±6, ±2, ±12, ±4

3 | -20 | +29 | +12 | -1 |
| -3 | +23 | -52 | |
3 | -23 | +52 | -40 | |
3 | -20 | +29 | +12 | 1 |
| 3 | -17 | +12 | |
3 | -17 | +12 | +24 | |

3 | -20 | +29 |+12 | 1/3 |
| 1 | -19/3 | +68/3 | |
3 | -19 | +68/3 | +176/9 | |
3 | -20 | +29 | +12 | -1/3 |
| -1 | +7 | -12 | |
3 | -21 | +36 | 0 | |

3x2-21x+36=0
x=-b±b2-4ac2a x=-(-21)±(-21)2-4(3)(36)2(3) x=21±36
x1=-1/3 | x2=3 | x3=4 |

2) 2x5-11x4+10x3+8x2=0

+ | - | i |
2 | 3 | 0 |
0 | 3 | 2 |
0 | 1 | 4 |
pq=±0, ±1, ±2, ±3, ±4, ±5…±1, ±2
pq=±0, ±0,±1, ±12,±2±1,±3, ±32,±4, ±2, ±5, ±52

2 | -11 | +10 | +8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | |
2 | -11 | +10 | +8 | 0 | 0 | |

2x4-11x3+10x2+8x+0=0

2 | -11 | +10 | +8 | +0 | 0 |
| 0 | 0 | 0 | 0 | |
2 | -11 | +10 | +8 | 0 | |

2x3-11x2+10x+8=0

2 | -11 | +10 | +8 | -1 |
| -2 | +13 | -23 | |
2 | -13 | +23 | -15 | |
2 | -11 | +10 | +8 | 1 |
| +2 | -9 | +1 | |
2 | -9 | +1| +9 | |


2 | -11 | +10 | +8 | 1/2 |
| 1 | -5 | +5/2 | |
2 | -10 | +5 | 21/2 | |
2 | -11 | +10 | +8 | -1/2 |
| -1 | +6 | -8 | |
2 | -12 | +16 | 0 | |


2x2-12x+16=0
x=-b±b2-4ac2a x=-(-12)±(-12)2-4(2)(16)2(2) x=12±44
x1=0 | x2=0 | x3=-1/2 | x4=2 | x5=4 |

3) x4+54x3-154x2-5x-1=0
4x4+54x3-154x2-5x-1=4(0)
4x4+5x3-15x2-20x-4=0

+ | - | i |
1 | 3| 0 |
1 | 1 | 2 |
pq= ±1, ±2, ±4±1, ±2, ±4
pq=±1, ±12,±14, ±2,±1±12, ±4, ±2,±1

4 | +5 | -15 | -20 | -4 | 1 |
| +4 | +9 | -6 | -26 | |
4 | +9 | -6 | -26 | -30 | |
4 | +5 | -15 | -20 | -4 | -1 |
| -4 | -1 | +16 | +4 | |
4 | +1 | -16 | -4 | 0 | |

4x3+x2-16x-4=0

4 | +1 | -16 | -4 | 1/2 |
| +2 | +3/2 | -29/4 | |
4 | +3 | -29/2 | -45/4 | |
4 | +1 | -16 | -4 | -1/2|
| -2 | +1/2 | +31/4 | |
4 | -1 | -31/2 | +15/4 | |


4 | +1 | -16 | -4 | 1/4 |
| +1 | +1/2 | -31/8 | |
4 | +2 | -31/2 | -63/8 | |
4 | +1 | -16 | -4 | -1/4 |
| -1 | 0 | +4 | |
4 | 0 | -16 | 0 | |


4x2-16=0
x=-b±b2-4ac2a x=-(0)±(0)2-4(4)(-16)2(4) x=0±168
x1=-1 | x2=-1/4 | x3=2 | x4=-2 |

4) x4-2x3-6x2-18x-135=0

+ | - | i |
1 | 3 | 0 |1 | 1 | 2 |
pq= ±1, ±3, ±5, ±9, ±15, ±27, ±45, ±135±1
pq=±1, ±3,±5, ±9,±15,±27, ±45, ±135

1 | -2 | -6 | -18 | -135 | 1 |
| +1 | -1 | -7 | -25 | |
1 | -1 | -7 | -25 | -160 | |
1 | -2 | -6 | -18 | -135 | -1 |
| -1 | +3 | +3 | +15 | |
1 | -3 | -3 | -15 | -120 | |

1 | -2 | -6 | -18 | -135 | 3 |
| +3 | +3 | -9 | -81 | |
1 | +1 | -3 | -27 | -216 | |
1 | -2 | -6 | -18 |-135 | -3 |
| -3 | +15 | -27 | +135 | |
1 | -5 | +9 | -45 | 0 | |

x3-5x2+9x-45=0

1 | -5 | +9 | -45 | 5 |
| +5 | 0 | +45 | |
1 | 0 | +9 | 0 | |
x2+9=0

x2+9=0
x=-b±b2-4ac2a x=-(0)±(0)2-4(1)(9)2(1) x=0±6i2
x1=-3 | x2=5 | x3=3i | x4=-3i |

5) x5+x4-14x3-2x2+24x=0
+ | - | i |
2 | 3 | 0 |
0 | 3 | 2 |
0 | 1 | 4 |
pq=±0, ±1, ±2, ±3, ±4, ±5…±1pq=±0, ±1, ±2, ±3, ±4, ±5…

1 | +1 | -14 | -2 | +24 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | |
1 | +1 | -14 | -2 | +24 | 0 | |

x4+x3-14x2-2x+24=0

1 | +1 | -14 | -2 | +24 | 1 |
| +1 | +2 | -12 | -14 | |
1 | +2 | -12 | -14 | +10 | |
1 | +1 | -14 | -2 | +24 | -1 |
| -1 | 0 | +14 | -12 | |
1 | 0 | -14 | +12 | +12 | |

1 | +1 | -14 | -2 | +24 | 2 |
| +2 | +6 | -16 | -36 | |1 | +3 | -8 | -18 | -12 | |
1 | +1 | -14 | -2 | +24 | -2 |
| -2 | +2 | +24 | -44 | |
1 | -1 | -12 | +22 | -20 | |

1 | +1 | -14 | -2 | +24 | 3 |
| +3 | +12 | -6 | -24 | |
1 | +4 | -2 | -8 | 0 | |

x3+4x2-2x-8=0

1 | +4 | -2 | -8 | 4 |
| +4 | +32 | +120 | |
1 | +8 | +30 | +112 | |
1 | +4 | -2 | -8 | -4 |
| -4 | 0 | +8 | |
1 | 0 | -2 | 0 | |

x2-2=0...