SECTION 1.6 FACTORING OUT THE GREATEST COMMON FACTOR

The greatest common factor (GCF) of a polynomial is the product of

  1. the GCF of the coefficients of all the terms of the polynomial and
  2. the variable factors common to all terms. (The exponent on each variable will be the smallest exponent that occurs on that variable factor in any of the terms.)

EXAMPLE 1         Factor out the GCF of 3x + 9.

                                    3 is the GCF.

                                  

EXAMPLE 2

Factor out the GCF of   
                                

The GCF of the coefficients is 10 (10 is the largest number that will divide into 30 and -10 and have no remainder.)

The GCF of the variables is   
                                        

 

So, we will factor out
                                   

Therefore,

                                   

EXAMPLE 3         Factor out the GCF of

                                                   

                                GCF: 

                               

 

 

FACTORING TRINOMIALS

A trinomial of the form is factorable into a pair of binomial factors with integer
coefficients if and only if there are two intgers whose product is c and whose sum is b.

 

SIGN PATTERN FOR THE FACTORS OF

If the constant c is positive, the factors of c must have the same sign. These factors will share the same sign as the linear coefficient b.

If the constant c is negative, the factors of c must be opposite in sign. The sign of the constant factor with the larger absolute value will be the same as that of the linear coefficient b.

Factor the following:

The question to ask yourself is "What factors of 8 add to 6?"
What factors of 18 add to -9?
What factors of -14 add to 5?
wpe1.jpg (1742 bytes) What factors of -40 add to -3?

 

TEST FOR FACTORING  

 
The trinomial with integer coefficients a, b and c will factor over the integers if
is not a perfect square, then is prime.

 

FACTORING TRINOMIALS BY SYSTEMATIC TRIAL AND ERROR

STEP 1 Make a form for the two binomial factors, and fill in the obvious information, such as the sign pattern.
STEP 2 List the possible factors fo the first and last terms that fit this pattern.
STEP 3 Select the factors that yield the correct middle term. If all possibilities fail, the polynomial is prime.

 

Factor
Sign pattern
Possible factors of Possible factors of -4
x and 6x 4 and -1
3x and 2x 2 and -2
1 and -4

 

So the possible ones are

This one is the correct factorization.

 

Factor
Sign Pattern

 

Possible Factors of Possible Factors of -35
3x and x 1 and -35
5 and -7
7 and -5
 

35 and -1

POSSIBLE FACTORS

This is the correct factorization.

NOTE: As soon as you get the right factorization, STOP. There is no need to continue listing all possible factors.