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.
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