Showing posts with label gcf. Show all posts
Showing posts with label gcf. Show all posts

Introduction to Factoring and Factor by Grouping

Factoring is one of the more important skills to learn in a beginning algebra course.

The idea is to work the distributive property in reverse. The goal is to write polynomials as products of simpler polynomials.

To decide what the GCF (Greatest Common Factor) is, look for the largest factor that divides into all the terms. In other words, what common term will divide into all the given terms evenly?
In the above example, the variable z is not common to all the terms so it is not included in the GCF. Use the smallest exponent for the common variables x and y.

Identify the GCF.
Factor out the GCF:

   Step 1: Identify the GCF.
   Step 2: Divide each term by the GCF to determine the remaining factor.
   Step 3: Check by multiplying.
The check is not really necessary; however, it is good practice to at least check the factoring mentally.

Factor the GCF out of the expression.
Tip: Be careful to use a 1 when the entire term factors out. This is where a mental check is important. Be sure that if you were to distribute you would get back to the original expression.

All of the above problems require only one step. The hard part is to identify the GCF. Here is the check for the last solved problem.
Factoring 4-Term Polynomials by Grouping
Now we will use the idea of factoring out the GCF in a technique called factoring by grouping of four-term polynomials. The steps are as follows.

   Step 1: Group the first two terms and the last two terms.  Factor out the GCF of both groupings.
   Step 2: If the remaining binomial factors are the same factor it out.
   Step 3: Check by multiplying.
Factor by grouping.

Sometimes you will encounter 4-term polynomials where factoring by grouping does not seem to work.  When this is the case, you can try to rearrange the terms in a different order and try again. For example, try to factor the following polynomial  by grouping.

Rearrange the terms and try again.
You can check by multiplying.
Video Examples on YouTube:

General Guidelines for Factoring Polynomials

Now that we have learned techniques for factoring 4, 3, and 2-term polynomials, we are ready to practice by mixing up the problems. The challenge is to first identify the type of factoring problem then decide which method to apply.

The basic guidelines for factoring follow:
  1. Look to factor out any GCF.
  2. Four-Term Polynomials - Factor by grouping.
  3. Trinomials - Factor using the "guess and check" method.
  4. Binomials - Use the the special products in this order:
Sum and Difference of Squares

Sum and Difference of Cubes

* If a binomial is both a difference of squares and cubes, then to obtain a more complete factorization, factor it as a difference of squares first.
* Not all polynomials factor.  In this case, beginning algebra students may write, "does not factor - DNF."

Tip: Make some note cards to aid in helping memorize the formulas for the special products. Look for factors to factor further - sometimes factoring once is not enough.

Take some time to understand the difference between the last two solved problems. Notice that x^6 - y^6  is both a difference of squares and a difference of cubes at the same time. Here we chose to apply the difference of squares formula first. On the other hand, for x^6 + y^6 we chose to apply the sum of cubes formula first because it does not factor as a sum of squares.


Video Examples on YouTube: Factor the following polynomials.