Associated Topics || Dr. Math Home || Search Dr. Math

### Solving Cubics (3rd Degree Polynomials)

```
Date: 12/15/96 at 11:28:34
From: Kenny Walden
Subject: Solving Cubics (3rd Degree poly's)

Is there a straightforward way to solve problems of the type
ax^3+bx^2+cx+d=0 ?  I know it's possible to do it guess and
check, but at competitions you don't have that much time.
A formula would be most helpful, but a step-by-step process of
factoring would also be very useful.
```

```
Date: 12/16/96 at 09:38:16
From: Doctor Jerry
Subject: Re: Solving Cubics (3rd Degree poly's)

Hi Kenny,

The formula for a cubic is much more complex than the formula for
solving a quadratic.  There is no known "step-by-step process of
factoring,"  other than trying the possible rational roots (for
polynomials with integer coefficients) by synthetic division.  That
is, if you want to solve the cubic:

2x^3+x^2+x-1=0

then (from a theorem), IF this equation has a rational root, it must
be in the list (1,-1,1/2,-1/2).  You can test these by synthetic
division.  Writing down 2 1 1 -1 and testing 1/2 gives the numbers
2 2 2 0.  The 0 means that 1/2 is a zero and that:

2x^3+x^2+x-1 = (x-1/2)(2x^2+2x+1).

If it happens that the polynomial equation has no rational roots, then
a numerical method or Cardan's Method must be used - for a complete
discussion, look at this answer in our archives:

http://mathforum.org/dr.math/problems/handel.6.20.96.html

-Doctor Jerry,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Polynomials

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search