Showing posts with label factoring. Show all posts
Showing posts with label factoring. Show all posts

Sunday, November 4, 2012

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.

Factoring Trinomials of the Form x^2 + bx + c

In this section we will factor trinomials - polynomials with three terms. Students find this difficult at first. However, with much practice factoring trinomials becomes routine. If a trinomial factors, then it will factor into the product of two binomials.

   Step 1: Factor the first term: x^2 = x*x.
   Step 2: Factor the last term. Choose factors that add or subtract to obtain the middle term.
   Step 3: Determine the signs by adding or subtracting the product of the inner and outer terms.
   Step 4: Check by multiplying.
Rather than trying all possible combinations of the factors that make up the last term spend some time looking at the factors before starting step two. Look for combinations that will produce the middle term. Here is the thought process in choosing 3 and 4 in step two above:
   "Can I add or subtract 1 and 12 to obtain 7?" – NO     
   "Can I add or subtract 2 and 6 to obtain 7?" – NO    
   "Can I add or subtract 3 and 4 to obtain 7?" – YES, because +3 + 4 = +7

Factor the trinomials.
This process used for factoring trinomials is sometimes called guess and check or trial and error. The biggest problem occurs when the signs are improperly chosen. With this in mind, you should take care to check your results by multiplying. Also, since multiplication is commutative order does not matter, in other words
If the trinomial has a GCF you should factor that out first.  Also, you should factor in such a way as to ensure a resulting trinomial with a positive leading coefficient.

Factor the trinomials.
Take care to perform the check. Most of the problems that you will encounter factor nicely but be sure to watch out for something like this . The middle term works but the last term does not:
because the sign of the last term is incorrect

Video Examples on YouTube: