Factoring Basics

Factoring Basics 

Factoring is a common math problem that people have asked about for centuries. There are many different types of factoring problems, but the most popular one is called the greatest common factor. This problem asks what number can be divided into both numbers without a remainder to make them equal? A GCF is required when solving most types of word problems in AP Calculus (AB). Of course, the best way to find a factor would be with a graphing calculator (TI-84 Plus CE), but this article will show you how you can use other simple techniques!

truefactors.org.

These sites are a great way to find a variety of facts and factors. 

http://www.mathforum.org/tools/methods/factoring/factoring.html 

Factoring is a natural process that occurs in all mathematics - including arithmetic, algebra, geometry and calculus - however it is primarily used to solve word problems in elementary algebra.

You may be able to use the following factoring algorithms:

Factoring Worksheets: http://www.teachersfirstmath.com/factoring-principles-worksheets.htm

Factoring with the TI-84 Plus CE: http://mathforum.org/toolbox/principles/calculus/factoring.html#PAPERS

Factors that are divisible by 5, or multiples of 5 are called perfect factors because they will always work in factorizing any number! Perfect factors for composite numbers include: 1, 4, 10, 15, 20...

The TI-84 Plus CE can factor numbers with the use of a combination lock. The key is to find a perfect factor using the least common multiple algorithm or by using trial and error. The least common multiple (LCM) counts all possible factors into which a number can be divided without a remainder. The key is to go through the numbers in increasing order until you find one that will always work are does not divide another number. If you get stuck, press in on the keys and see what happens. In this way you can find a factor for any number.

Simply put, the LCM is the product of all the common multiples in that range. For example the least common multiple of 5, 9, and 11 is 25. The numbers 1, 2, 4 and 8 will all work here because 1 only divides into 2, 4 and 8 without any remainder. The least common multiple of 20 is also 25 so be sure to check your numbers as you factor! Look at how many factors 1 has, how many factors 4 has etc until you find one that works perfectly!

If finding perfect factors takes too long (or an incorrect number is found) you may have to resort to trial and error. It will work nearly every time!

Trial and Error: 

This method is the most accurate way to find a factor when using the TI-84 Plus CE. First press . Next, enter in the number to be factored. Press , then , and finally again. If you get an error, then try another number that is greater or less than your original number by one. This will continue until you find a viable factorization.

Since many students are familiar with the process of factoring trinomials, it is important for them to recognize that finding factors for polynomials is not as complicated as it may first seem. All polynomials can be factored in one of two ways. The first is to factor the trinomial into three separate binomials that may be more easily worked with.

If the trinomial has a term in the numerator that is a perfect square, then those terms are distributed over the binomials and then everything is factored by getting the GCF. This makes it easier to complete the division by using factors of two and three and will lead you to an answer that is easier to work with.

The first step is to clear the fractions from the trinomial and factor out the GCF if necessary. The most important thing to remember is that you must regroup when you have several like terms in a binomial (like 2x + 4x). If your polynomials are complex, you may need to simplify them by combining like terms in order to get a result that is easier to work with.

Example:  (2x + 3)(2x + 7) 

Start by distributing 2x over both of the binomials and then factoring out the GCF of 3. Therefore the answer would be (2x)(x + 7)

Example:  (5a + 6)(5a - 9) 

The first step is to clear the fractions by multiplying them out until you get an expression that can be distributed over each term in the first factor and a binomial in the second factor.

This will result in (5a + 6)(5a − 3). The next step is to take out the GCF of 9 from both factors. If you have some terms that are like terms, then they must be regrouped so that they are easier to work with. This will result in 5ax + 3ax = 2(2ax + 3a). This factor also works because 2(2a + 3) = 23, so this is the final answer. 

For trinomials, there are many different ways to factor them. The most important thing to remember when factoring a trinomial is that you first have to factor out any GCF or GCF's and then distribute the factors evenly over each part of the trinomial. You can use trial and error methods until you find a way in which your numbers work, but it is much quicker and easier to do this with a graphing calculator.

Example:  (x - 2)(x + 3)

This trinomial has a GCF of x - 2 and should be distributed over each part of the trinomial. This will result in (x)(x - 6). The next step is to distribute the 3 over both parts of the binomials so that each one can be then factored with a GCF or by using trial and error.

Example:  (3y - 4)(3y + 7) 

The first step is to distribute the three over one of the terms and then use trial and error to find another factor for each of the numbers in the other part of this trinomial.

Conclusion: 

By distributing the GCF's of the polynomials you can factor them out in many different ways. The most important thing to remember is that you have to first regroup terms that are like terms and then factor them out by using trial and error until you find a way in which your numbers work!

In algebra, there are two rules that can be used to factor polynomials. In the following examples the term will be assumed to be monic, so we assume y=x+a.