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 Depdendent Variable

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 Dependent Variable

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# Factoring

Definition

To factor a polynomial is to find a product of polynomials such that the product is equivalent to the original polynomial.

• Factoring Monomials.
• 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.