Factoring Part 1

Factoring polynomial expressions is likely a new skill, or at least one you have little experience with.  Factoring means to break something down into its multiplicative parts, to un-distribute.  In other words, when you have a polynomial and you’re trying to factor it, you’re trying to figure out what small polynomials have a product that is the current expression.

Factoring is a bit of a gate-way skill.  Without the ability to factor correctly, quickly, and consistently, moving forward in mathematics will become increasingly difficult.  As such, spend time on the topic.  Several pages will be devoted to factoring on this site.  There will be a LOT of videos.  

Give yourself the time and practice to learn this.  The investment in yourself will pay off!  Remember, when opportunity presents itself, it is too late to prepare.  One of the things you do with education is prepare for unseen, future, opportunities. 

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There are four cases of factoring that we will eventually learn.  They are as follows:

  1. Factoring out a constant term (monomial)
  2. Factoring a quadratic when a = 1
    1. What multiplies to c and adds to b?
  3. Factoring a quadratic when a ≠ 1
    1. Guess and Check Method
  4. Factor by Grouping

The first method, factoring out a constant is, is something you should always try.  It is always a valid approach.  The other methods are contextually specific.  But, before getting into all of that, let’s discuss the basics first.

Factoring is the opposite of distributing.  If you factored the number 72, you might say 8 × 9.  It is no different with polynomials.  For example, you could factor 3x + 6 as the product of 3 and the binomial x + 2.  It would be written as follows:

3x + 6 = 3(x + 2)

This is an example of factoring out a monomial, or a single term.  Before getting into that, there are a handful of things you must understand about factoring.

  1. What is factoring?
  2. How to factor?
  3. When to factor?
  4. If the factoring is correct?
  5. If the factoring is complete?

 

Factoring is breaking a polynomial down to its smallest multiplicative parts.  You should factor whenever it suits the situation.  For example, if you’re stuck, or do not see how to simplify something, often factoring unlocks the path to clarity.  This is so often the case that we can joke that the acronym WTF, which would be spoken when frustrated, stands for What the Factor, in math class.  The reason is that if you’re stuck and frustrated from this point forward, you should try to factor.

The reason factoring is such a great tool is that it can take large, difficult to handle expression, and break it down into chunks that are easier to manipulate and understand.

You’ll know when you’re done factoring because you cannot break down the polynomial any further.  Consider the number 72 that we factored early into the product of 8 and 9.  This is a valid factorization, it is correct because 8 × 9 = 72, but it is not completely factored because it can be further broken into multiplicative parts.  The same is true for factoring polynomials.  You’ll learn more advanced factoring techniques as you move through this section, and so your understanding of what is fully factored, or not, will change as you learn more.

Method #1

Factoring a Constant

The first thing to examine when factoring is if each term in a polynomial shares a common factor.  If it does, and you can divide out that greatest common factor (from all terms), you’ll have factored that polynomial.

18x2y + 27xy2 – 3xy

This trinomial has a common factor of 3xy.  Each term shares that factor, but not all three terms share a larger factor.  The first two terms share a factor of 9, but the last term does not have a factor of 9.  So, we can divide 3xy from each term.

This is correct because if we distribute the 3xy to all terms, we end up with the original expression as our product.  We are done because we cannot further factor the remaining terms.

Prime polynomials cannot be factored, save by one and itself.  6x + 9y – 1 is considered a prime polynomial, so we are done factoring because we have that multiplied with a monomial.

To factor out a constant you need to identify if a greatest common factor exists between all terms of the polynomial.  Then, divide it out, and write the polynomial as the product of the GCF and the remaining values.

Example:  Factor 5x2 – 25x – 10

Here the GCF of all terms is just 5.  That’s the largest number that divides into all terms.  If we divide out five we get the following.

5(x2 – 5x – 2)