aх^3 + bх^2 + cx + d = 0 is a generalized form of a cubic equation.
The conventional method for solving a cubic equation is to convert it to a quadratic equation and then solve it using factoring or the quadratic formula. A cubic equation always has at least one actual root, unlike a quadratic equation, which may have no genuine solution.
Determine the roots of the cubic equation
2х^3 + 3х^2 – 11x – 6 = 0
The possible factors are 1, 2, 3, and 6 (since d is 6).
Here, you need to apply the Factor Theorem to check the possible values
f (1) = 2 + 3 – 11 – 6 ≠ 0
f (–1) = –2 + 3 + 11 – 6 ≠ 0
f (2) = 16 + 12 – 22 – 6 = 0
So, x = 2 is the first root.
We can get the other roots of the equation using the synthetic division method.
= (x – 2) (ax2 + bx + c)
= (x – 2) (2x2 + bx + 3)
= (x – 2) (2x2 + 7x + 3)
= (x – 2) (2x + 1) (x +3)
Answer: x = 2, x = -(1/2), and x = -3