Finding Factors of Polynomials
Finding factors of polynomials is important, since it is always best to work with the simplest version of a polynomial.
Learning Objective

Practice the different methods for finding the factors of a polynomial
Key Points
 Factoring is a critical skill in simplifying functions and solving equations.
 There are four types of factoring shown, which are 1) "pulling out" common factors, 2) factoring perfect squares, 3) the difference between two squares, and then 4) how to factor when the other three techniques are not applicable.
 The first step should always be "pulling out" common factors. Even if this does not factor out the polynomial completely, this will make the rest of the process much easier.
Terms

common factor
A value, variable or combination of the two that is common to all terms of a polynomial.

factor
To express a mathematical quantity as a product of two or more like quantities.
Full Text
When multiplying, things are put together. When factoring, things are pulled apart. Factoring is a critical skill in simplifying functions and solving equations.
There are four basic types of factoring. In each case, it is beneficial to start by showing a multiplication problem, and then show how to use factoring to reverse the results of that multiplication.
"Pulling Out" Common Factors
This type of factoring is based on the distributive property, which states:
When factoring, this property is done in reverse. Therefore, starting with an expression such as the one above, it can be noted that every one of those terms is divisible by
We now divide each term with this common factor to fill in the blanks. For instance,
For many types of problems, it is easier to work with this factored form.
As another example, consider
There are two key points to understand about this kind of factoring:
 This is the simplest kind of factoring. Whenever trying to factor a complicated expression, always begin by looking for common factors that can be pulled out.
 The factor must be common to all the terms. For instance,
$8x^314x^2+6x+7$ 8 has no common factor, since the last term,$7$ , is not divisible by$2$ or$x$ .
Perfect Squares
The second type of factoring is based on the "squaring" formulae:
For instance, if the problem is
If the middle term is negative, then the second formula is:
This type of factoring only works in this specific case: the middle number is something doubled, and the last number is that same value squared. Furthermore, although the middle term can be either positive or negative, the last term cannot be negative. This is because if a negative is squared, the answer is positive.
To use this method of factoring, one must keep their eyes open to recognize the pattern. The best way to do this is practice.
Difference Between Two Squares
The third type of factoring is based on the third of the basic formulae:
This formula can be run in reverse whenever subtracting two perfect squares. For instance, if there is
Note that, in the last example, the first step is done by pulling out a factor
Note also that when we are working with real numbers, all positive numbers are squares. So
It often happens that we can use this method twice (or more):
It is important to note that the sum of two squares cannot be factored.
As in the case of factoring a perfect square, to use this method one has to keep their eyes open to notice the pattern.
Brute Force Factoring
This is the hardest way to factor a polynomial, but the one we need to use when the other ones do not suffice.
In general, we can multiply any number of polynomials with any number of terms using the distributive property.
To see how to use this for factoring, we again try to notice a pattern. For example, we have:
Since
In general:
Or, of course:
which we can factor again by the previous method if
Especially when we think
Key Term Reference
 difference
 Appears in these related concepts: Arithmetic Sequences, Asymptotes, and The General Term of a Sequence
 equation
 Appears in these related concepts: A General Approach, Equations and Inequalities, and What is an Equation?
 expression
 Appears in these related concepts: Bacterial Transformation, Graphs of Equations as Graphs of Solutions, and Simplifying, Multiplying, and Dividing Rational Expressions
 factoring
 Appears in these related concepts: Selling to Consumers, Rational Algebraic Expressions, and Factoring Accounts Receivable
 function
 Appears in these related concepts: Functions and Their Notation, The Vertical Line Test, and What is a Linear Function?
 integer
 Appears in these related concepts: Binomial Expansions and Pascal's Triangle, Scientific Notation, and Finding a Specific Term
 point
 Appears in these related concepts: The Intermediate Value Theorem, Polynomial and Rational Functions as Models, and Graphing Equations
 polynomial
 Appears in these related concepts: Domains of Rational and Radical Functions, Finding Polynomials with Given Zeros, and Counting Rules and Techniques
 real number
 Appears in these related concepts: Introduction to Complex Numbers, Solving Problems with Inequalities, and Zeros of Polynomial Functions with Real Coefficients
 real numbers
 Appears in these related concepts: Piecewise Functions, Introduction to Domain and Range, and Linear Inequalities
 square
 Appears in these related concepts: Radical Equations, Factoring a Difference of Squares, and Matrix Multiplication
 sum
 Appears in these related concepts: Scientific Applications of Quadratic Functions, The Order of Operations, and Basic Operations
 term
 Appears in these related concepts: Basics of Graphing Polynomial Functions, Introduction to Variables, and Identify and Simplified Like Terms (MLS 2.1)
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