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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: 


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:   

I hope this answers your question.

-Doctor Jerry,  The Math Forum
 Check out our web site!   
Associated Topics:
High School Polynomials

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