Cubic formula

Consider the cubic equation ax^3 + bx^2 + cx + d = 0.

If Δ < 0, then the equation has three distinct real roots.


Case 1 (3 fractional or integral roots),

the equation can be written as:

[(a1)x - (b1)][(a2)x - (b2)][(a3)x - (b3)] = 0

where a1, a2, a3, b1, b2 and b3, are all integers.

You can try all the factors of a and d, including positive and negative ones, then you will find a1, a2, a3 and b1, b2, b3 are some of the factors of a and d respectively.


Case 2 (one of the three real roots is fraction or integer),

the equation can be written as:

(px - q)(hx^2 + kx + m) = 0

where p, q, h, k and m are all integers.

Then, by using quadratic formula, you can solve hx^2 + kx + m = 0.


Case 3 (none of the three real roots is fraction or integer),

you may try the numerical method such as Newton's method. However, the solutions are just approximations.

Let f(x) = ax^3 + bx^2 + cx + d, then try to get when f(x) is equal to 0.

For more information, you may look for the materials about Newton's method.


Last question: How do I know what are the nature of roots before finding them out?

Answer: Yes, you don't know. Therefore, you should try all the integral and fractional solutions first. If it does not work, then try the numerical method.

http://simonyau2000.net/mathspace/algebra.pdf
此網有介紹三次方程及四次方程