Showing posts with label factor. Show all posts
Showing posts with label factor. Show all posts

Sunday, November 4, 2012

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

So far we have looked at trinomials with leading coefficients of 1. Now we will see how the process is changed when the leading coefficients are something other than 1.

   Step 1: Look at the factors of the first and last terms.
   Step 2: Choose the factors where the multiples add to the middle term.
   Step 3: Determine the signs using the product of the inner and outer terms.
   Step 4. Check by multiplying.

As you can see there are many more combinations to consider when the leading coefficient is not 1. These take time but become routine with practice.

Factor the trinomials.
Video Examples on YouTube:

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:

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: