Use the Simplify > Factor commandU08MH3 or the Factoring FunctionsFactoring_Functions to factor a polynomial with respect to some or all of its variables.


The general objective when factoring a polynomial with respect to designated variables is to maximize the number of independent factors in which the designated variables occur linearly and which have the simplest possible coefficients.  There is often a tradeoff between these two objectives.  For example, linear factors might be achievable only at the expense of complex coefficients.  The following summarizes the types of polynomial factoring:


•        If trivial content factoring a polynomial, it is placed over a common denominator, and the gcd of the numeric coefficients and the least powers of variables of terms are factored out.  For example,

6·x^2 + 10·x^3/y

trivial content factors to

    2              
 2·x ·(5·x + 3·y)  
————————————————— 
         y         


•        If squarefree factoring a polynomial, in addition to trivial content factoring, powers of sums or products of different powers of sums are factored.  For example,

x^4 + 2·x^3 - 3·x^2 - 8·x – 4

squarefree factors to

       2   2      
(x + 1) ·(x  - 4) 

Note that x²−4 was not factored since x+2 and x−2 are the same power (i.e. the first power) of sums.


•        If rationally factoring a polynomial, in addition to squarefree factoring, products of sums are factored so long as no new fractional powers or complex numbers are introduced.  For example, the expression above rationally factors to

                       2 
(x + 2)·(x - 2)·(x + 1)  


•        If radically factoring a polynomial, in addition to rational factoring, the possibility of introducing fractional powers of numbers, such as √2, is allowed.  For example,

x^2 - 3

radically factors to

(x + √3)·(x - √3)


•        If complex factoring a polynomial, in addition to radical factoring, the possibility of introducing complex numbers is allowed.  For example,

x^2 + 3

radically factors to

(x + √3·i)·(x - √3·i)


Fractional powers of complex numbers may introduce trigonometric and inverse trigonometric functions to achieve normal rectangular formRectangular_form.  In approximate or mixed mode, such subexpressions simplify to approximate real or complex numbers for numeric arguments.  In exact mode, they are left in symbolic form, which may be quite large.


Use radical factoring if you want real factors that are factored as far as possible.  Use complex factoring if you want factors that are linear factors in a variable.  Do not use radical or complex factoring if you want a concise expression without new fractional powers and/or complex numbers.


Subexpressions that do not contain factorization variables are simplified without being unnecessarily transformed.  In other words, subexpressions involving only non-factoring variables are neither expanded nor factored unnecessarily.  Similar powers of factoring variables are collected and simplified.  New radicals introduced by radical or complex factoring may include non-factorization variables as well as numbers.


Factoring a polynomial may take a long time depending on the size of the expression, the type of factoring, and the number of factorization variables.  Click on the Calculation Progress dialog box's Abort button or press the Esc key to abort an Simplify > Factor command that is taking too long (for details, see the Simplify > Basic commandBQ.NIL).  


Factoring only subexpressions of an expression or factoring with respect to a different variable may drastically reduce the computing time and improve the comprehensibility of a result.

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