Factoring

To factor a polynomial means to find the factors (numbers that multiply) which form the polynomial product.  The first factor that you should look for is the greatest common factor.  The GCF is the largest number that will evenly divide the terms of a given polynomial.  In essence, factoring a GCF is a ‘reversal’ of the distributive property.  For example, if you distribute 3(4x – 5) = 12x – 15.

Consider the binomial 12x – 15.  To factor the GCF, determine what is the largest number that will evenly divide both 12x  and  –15?  Since 3 is the GCF, divide both 12x and  –15 by 3 and you get the factored form of the polynomial.  Thus, 12x – 15 = 3(4x – 5).

Another lengthier example is.

is the GCF since it is the largest number that will evenly divide all of the terms of the polynomial.  Divide the terms of the polynomial  by  and you will get the factored form which is   The terms of the factor no longer have any common factor; therefore, we say that the polynomial is fully factored.

Next, once you have factored the GCF (if there is a GCF in the polynomial), consider the length of the polynomial.  How many terms are in the polynomial?  If there are four terms, try factoring by grouping.  Factor by grouping involves pairing the four factors into groups of two terms, factoring a GCF from each of the pairs, and then factoring the GCF from the resulting pair of terms.  In essence, this method is the ‘reversal’ of foiling two binomials.

For example, if we multiply .
Now factor, , which is a four term polynomial.  The GCF between the first two terms  is  and the GCF between the second two terms  is .  Factoring you would get, .  Now factor the GCF which isbetween the terms  and  and you would get the factored form of the four-termed polynomial which is  … which by the commutative property is .  In other words, the order of the factors does not matter.  The two answers are equivalent.  Notice that the sign (+) of the 3^rd term is factored.  Always factor the sign of the third term.  This is important and can be tricky if the third term is negative.

If the polynomial has three terms, then try one of the following methods.  If the trinomial is ‘simple’, i.e. begins with a squared variable term with a coefficient of 1, then factor by the following method … which again is a ‘reversal’ of foiling.  Consider the following four examples:

1)   = (x      )(x      ) … to find the two missing numbers, including their signs, think of two number that you would multiply and get the last term +6 and at the same time ADD to get the middle term +5x.  These two numbers must be +3 and +2, since (+3)(+2) = +6 and  +3 +2 = +5.  Therefore the factored problem becomes,

2)   = (x      )(x      ) … to find the two missing numbers, including their signs, think of two number that you would multiply and get the last term +12 and at the same time ADD to get the middle term –7x.  These two numbers must be –3 and – 4, since (–3)( – 4) = +12 and  –3 – 4 = –7.  Therefore the factored problem becomes,

3)   = (x      )(x      ) … to find the two missing numbers, including their signs, think of two number that you would multiply and get the last term –8 and at the same time SUBTRACT to get the middle term  +2x.  These two numbers must be +4 and –2 since (+4)( –2) = –8 and  +4 –2 = +2.  Therefore the factored problem becomes,

4)   = (x      )(x      ) … to find the two missing numbers, including their signs, think of two number that you would multiply and get the last term –6 and at the same time SUBTRACT to get the middle term  –3.  These two numbers must be +2 and –3 since (+2)( –3) = –6 and  +2 –3 = –1.  Therefore the factored problem becomes,

If the trinomial is ‘complicated’ by having a coefficient in front of the squared variable term, then you may want to try the method of factoring trinomials by "re"grouping.  In this method, a trinomial is transformed into a four-termed polynomial and then factored by grouping as described above.
Consider factoring the trinomial .  If the coefficient of the first term (8) is multiplied times the last term (–3) the result would be  –24.  Think of two numbers that would multiply to give you –24 and at the same time SUBTRACT to give you the coefficient of the middle term –10x (using the same thoughts as were discussed above with the 'simpler' trinomials.)  The numbers are +2 and –12, since (+2)(–12) = –24 and  +2  –12 =  –10.  Next, take that information and rewrite the middle term, since , you can substitute.  This would now look like:  Now, take  and factor by grouping as described above and the result would be

If the polynomial has two terms, then the binomial may either be the difference of squares or the sum or difference of cubes.  The difference of squares has to be recognized by observing the first term to be a squared number and the last term to be a squared number.  To factor the difference of squares,  take the square root of the first term and then the square root of the second term.  The square root of
 and the square root of 25 = 5, now insert as follows:   
Notice, the result is a pair of conjugates … which means two factors that look alike except that one has a + and one has a – connecting the terms of the binomials.   Note:  you cannot factor the sum of squares.

To factor the sum or difference of cubes, you must know that the result is a binomial factor and a trinomial factor.  Also, you must memorize the sign pattern.  The pattern is easily memorized by using the mnemonic SOAP.  S stands for Same (the first sign is the same as the binomial being factored),
O stands for Opposite (the second sign is the opposite of the binomial being factored), and AP stands for Always Positive (the last sign is always positive.)
To factor , which is the difference of cubes, first write down the sign pattern …
(  __  –  __  )(  __  +  __  +  __  ).  Find the cube roots of both of the terms of the polynomial.
The cube root of  and the cube root of 27 = 3.  Next, fill in the blanks by placing 2x in the first blank and 3 in the second blank.  Square 2x to get  for the third blank, multiply (2x)(3) to get 6x for the fourth blank and square 3 to get 9 for the last blank.  The result would look like this: .  The sum of cubes,  , would be the same as above, except for the signs, (  __  +  __  )(  __  –  __  +  __  ).  Since  and , then

The last step in factoring a polynomial is considering the question, Can any factors be factored further?  This step is a "last-check" in case you have not factored a GCF, or in case you can one of factors further.  Make sure that all of the factors in your answer are factored completely.

Practice is the key to any of these factoring methods!