   SEARCH HOME Math Central Quandaries & Queries  Question from Pedro, a student: I am having trouble figuring out the solutions for the polynomial equation 6x^3-17x^2+4x+12=0 Pedro,

I don't blame you; while there is a "cubic formula" that will grind out the solutions, it's a bit of a bear and most people try to avoid using it.

I suggest the following.

1. Plot the polynomial for a few easy values, say

-3, -2, …, 3

(plus-or-minus 3 because 3 is about 17/6, beyond which the cubic term will probably dominate, as the linear & constant terms are not big.)

2. If you hit a root a, divide out x-a and use the quadratic formula on what if left. [If you have a remainder you made a mistake]

3. Otherwise if p(n) and p(n+1) are on opposite sides of zero try simple fraction values between n and n+1. Or obtain an accurate graph using a graphing calculator and guess the zeros.

(1) Plot the polynomial for a few easy values, say

Other observations (think about these carefully)

*There is a negative root, because
p(-N)<0 for big N, p(0) > 0, and p(N)>0. Thus the graph crosses the x axis between -N and 0. If there is only one real root this is it.

* The sum of the roots is 17/6 and the product is -12/6. [Why? Expand (x-a)(x-b)(x-c) and look at the coefficients!] Thus if there are three real roots two are positive. This might make positive values a good place to start looking.

Good Hunting!
RD     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.