Find the roots of the polynomial equation
X^3-2x^2+10x+136=0
To find the roots of the polynomial equation x^3 - 2x^2 + 10x + 136 = 0, we can use various methods such as factoring, synthetic division, or the rational root theorem.
If we observe the coefficients of the equation, we can see that it doesn't seem to be easily factorable or have any obvious rational roots. Therefore, let's try using synthetic division to test for possible rational roots.
By the rational root theorem, the possible rational roots are the factors of the constant term (136) divided by the factors of the leading coefficient (1).
The factors of 136 are: ±1, ±2, ±4, ±8, ±17, ±34, ±68, ±136.
The factors of 1 are: ±1.
Using synthetic division with the possible rational roots, we can test each of these values until we find a remainder of 0.
Trying x = 1:
1 | 1 -2 10 136
|___ 1 -1 9
-----------------
1 -3 9
The remainder is not 0.
Trying x = -1:
-1 | 1 -2 10 136
|___ -1 3 -13
------------------
1 -3 -3
The remainder is not 0.
Trying x = 2:
2 | 1 -2 10 136
|___ 2 0 20
-----------------
1 0 30
The remainder is not 0.
Trying x = -2:
-2 | 1 -2 10 136
|___ -2 8 -36
------------------
1 -4 -26
The remainder is not 0.
Trying x = 4:
4 | 1 -2 10 136
|___ 4 8 72
-----------------
1 2 82
The remainder is not 0.
Continuing this process for the remaining rational roots, we find that none of them result in a remainder of 0. This means that there are no rational roots for the given polynomial equation.
Therefore, the polynomial equation x^3 - 2x^2 + 10x + 136 = 0 does not have any rational roots. The roots of this equation may be irrational or complex numbers.