Find all the roots of the equation
x^4 - 4x^3 + x^2 + 12x -12 = 0
To find the roots of the equation, we can use factoring, synthetic division, or the quadratic formula. Instead of solving the equation directly, we will first try to identify potential rational roots using the rational root theorem. According to the rational root theorem, the potential rational roots are the divisors of the constant term (-12) divided by the divisors of the leading coefficient (1).
The potential rational roots are ±1, ±2, ±3, ±4, ±6, and ±12.
Next, we will try these potential roots using synthetic division.
When x = 1:
1 | 1 -4 1 12 -12
| 1 -3 -2 10
---------------
1 -3 -2 10 -2
When x = -1:
-1 | 1 -4 1 12 -12
| -1 5 -6 -6
---------------
1 -5 6 6 -18
When x = 2:
2 | 1 -4 1 12 -12
| 2 -4 -6 -12
----------------
1 -2 -3 6 -24
When x = -2:
-2 | 1 -4 1 12 -12
| -2 12 -26 28
-----------------
1 -6 13 -14 16
When x = 3:
3 | 1 -4 1 12 -12
| 3 -3 -6 -18
----------------
1 -1 -2 6 -30
When x = -3:
-3 | 1 -4 1 12 -12
| -3 21 -66 126
-----------------
1 -7 22 0 114
When x = 4:
4 | 1 -4 1 12 -12
| 4 0 4 64
----------------
1 0 5 16 52
When x = -4:
-4 | 1 -4 1 12 -12
| -4 32 -124 472
------------------
1 -8 33 -112 460
When x = 6:
6 | 1 -4 1 12 -12
| 6 12 78 540
-----------------
1 2 13 90 528
When x = -6:
-6 | 1 -4 1 12 -12
| -6 60 -366 2220
------------------
1 -10 61 -354 2208
When x = 12:
12 | 1 -4 1 12 -12
| 12 96 1188
------------------
1 8 1092 1176
When x = -12:
-12 | 1 -4 1 12 -12
| -12 192 -2736
-------------------
1 -16 193 -2724
From the synthetic divisions, we found that none of the potential rational roots (±1, ±2, ±3, ±4, ±6, ±12) are actual roots of the equation. Therefore, the equation x^4 - 4x^3 + x^2 + 12x -12 = 0 does not have any rational roots.
To find the irrational or complex roots, we can use a computer or calculator to estimate the roots. Alternatively, we can use numerical methods such as the Newton-Raphson method or the bisection method.