Factoring
Definition
To factor a polynomial is to find a product
of polynomials such that the product is equivalent to the
original polynomial.
 Any given monomial can be factored a number of ways.
Example:
Factor 24x^{ 2} four different ways.
 A complete factorization of a monomial involves factoring
the coefficient into primes and leaving the variable(s)
alone.
Example:
Factor 24x^{ 2} completely.
 “Factoring out” a common factor.
Definition
A common factor for a collection of terms is
any number, variable, or algebraic expression which is a factor
of each of those terms.
Example:
Find all common factors of the terms 6x^{ 2} and 12x^{
14}.
Definition
The greatest common factor for a collection
of terms is the product of the largest number which is a common
factor of the terms and the highest power of each variable which
is a common factor of the terms.
Example:
Find the greatest common factor of the terms listed in the
last example.
 Look at what happens when we distribute something.
 Removing (or “factoring out”) the greatest
common factor is simply “undistributing” it.
Example:
Factor out the greatest common factor.
6x^{ 2} + 12x^{ 4}
 If the leading coefficient of a polynomial is negative,
factor out 1.
ALWAYS FACTOR OUT THE GREATEST COMMON FACTOR BEFORE
DOING ANY OTHER FACTORING!
 Common factors are not necessarily monomials.
Example:
4a^{ 2} ( 2a  1)  5(2a  1)
One method for factoring polynomials with four terms, called factoring
by grouping uses the idea of common factors which are
not monomials.
