find the roots of the polynomial equation
x^3-2x^2+10x+136=0
A) -3 + 5i, -4
B) 3+ i, 4
C) 3+5i,-4
D) -3+i, 4
To find the roots of the polynomial equation, we can use the Rational Root Theorem or synthetic division. After trying various options, we can see that -4 is a root of the equation.
Using synthetic division, we divide the polynomial by (x + 4):
-4 | 1 -2 10 136
| -4 24 -136
----------------------
1 -6 34 0
The remainder is zero, so (x + 4) is a factor of the polynomial. The resulting quadratic equation is x^2 - 6x + 34 = 0.
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / (2a)
In this case, a = 1, b = -6, and c = 34. Plugging these values into the quadratic formula, we get:
x = (-(-6) ± sqrt((-6)^2 - 4(1)(34))) / (2(1))
= (6 ± sqrt(36 - 136)) / 2
= (6 ± sqrt(-100)) / 2
= (6 ± 10i) / 2
= 3 ± 5i
Therefore, the roots of the polynomial equation x^3 - 2x^2 + 10x + 136 = 0 are 3 + 5i, 3 - 5i, and -4.
The correct answer choice is:
C) 3 + 5i, -4