Example Group #1 

Factoring out the Greatest Common Factor (GCF) is perhaps the most used type of factoring because it occurs as part of the process of factoring other types of products. Before you can factor trinomials, for example, you should check for any GCF. 



Example Group #2 

Oftentimes when there is no factor common to all terms of a polynomial there will be factors common to some of the terms. A second technique of factoring called grouping is illustrated in the following examples. 



Example Group #4 

Factoring the sum or difference in two perfect cubes is our next technique. As with squares, the difference in two cubes means that there will be two terms and each will contain perfect cubes and the sign between the two terms will be negative. The sum of two cubes would, of course, contain a plus sign between the two perfect cube terms. The follow formulas are helpful for factoring cubes: Sum: Difference: Notice that the sum and the difference are exactly the same except for the signs in the factors. Many students have found the acronym SOAP extremely helpful for remembering the arrangement of the signs. S represents the fact that the sign between the two terms in the binomial portion of the answer will always be the same as the sign in the given problem.
O implies that the sign between the first two terms of the trinomial portion of the answer will be the opposite of the sign in the problem.
AP states that the sign between the final two terms in the trinomial will be always positive. 



Example Group #5 

Before factoring a trinomial, examine the trinomial to be sure that terms are arranged in descending order. Most of the time trinomials factor to two binomials in product form. 



Example Group #6 

A general trinomial is one whose first term has a coefficient that can not be factored out as a GCF. The method of trial and error will be used to mentally determine the factors that satisfy the trinomial. We will show you the steps to factor each of the following general trinomials completely. 


#16: Factor the following problem completely.
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 In factoring the general trinomial, begin with the factors of 12. These include the following: 1, 12, 2, 6, 3, 4. As a general rule, the set of factors closest together on a number line should be tried first as possible factors for the trinomial.
 The only factors of the last term of the trinomial are 1 and 3, so there are not other choices to try. Because the last term is negative the signs of the factors 1 and 3 must be opposite.
 This is the first trial. The answer must be checked by multiplication, as follows:
 The factorization of the trinomial is almost correct. However, the sign of the middle term is incorrect. That means that the signs of the two factors should be switched.





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#17: Factor the following problem completely.
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 Factor out the negative sign first. Doing so will change all the signs of the trinomial.
 Now factor the trinomial. Factors of the first term include 1, 4, 2. Factors of the last term include 1, 6, 2, 3. The sign of the 6 is negative, so the signs in the two factors must be opposites.
 Consider 2 and 2 as factors of 4, and 3 and 2 as factors of 6.
 Such choices are not good, because it causes the second factor to contain a GCF and that should be avoided. A second attempt must be made, since checking the factors will fail as follows:
 Trials would continue by perhaps trying to switch the 3 and 2; however, that would cause a GCF in the first set of parentheses. That should be avoided, so the next idea would be to use 6 and 1 instead of 3 and 2.





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