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Int. Alg. Notes
Section 5.4
Page 1 of 3
Section 5.4: Greatest Common Factor; Factoring by Grouping
Big Idea: Polynomials are the most important topic in algebra because any equation that can be written using
addition, subtraction, multiplication, division, integer powers, or roots (which are rational powers) can be
solved by converting the equation into a polynomial equation. The fourth step toward acquiring this awesome
power is to be able to factor polynomials, which is simply “undoing” multiplication of polynomials.
Big Skill: You should be able to factor polynomials by factoring the greatest common factor and by factoring
by grouping.
In arithmetic, when you are asked to factor a number, you are supposed to find numbers whose product is the
number you are factoring.
Example:

Factor 24:
24 = 2  2  2  3

The above is an example of a prime factorization of 24 because all of the factors are prime. We can also
say 24 has been factored completely.
In algebra, when you are asked to factor a polynomial, you are supposed to find other polynomials whose
product is the polynomial you are factoring.
Examples:

Factor x 2  2 x  1 :
x2  2 x  1   x  1 x  1

Factor x3  3x 2  3x  1 :

Factor x 3  1 :
x3  3x2  3x  1   x  1 x  1 x  1
x3  1   x  1  x 2  x  1

The above examples are prime factorizations because each of the factors are prime polynomials. We say
that the polynomials above have been factored completely.
In arithmetic, an integer is prime when it can not be written as the product of integers other than itself and 1.
In algebra, a polynomial with integer coefficients is prime when it can not be written as the product of two
other polynomials with integer coefficients (other than itself and 1).
Examples:

7 is prime

 x  1 is prime

x
2
 x  1 is also prime
Our goal in the next three sections is to learn how to completely factor any given polynomial, and there are a lot
of techniques.
We begin with the simplest technique, factoring the greatest common factor.
Factoring the greatest common factor:
1. Identify the greatest common factor (GCF) in each term.
2. Rewrite each term as the product of the GCF and the remaining factor.
3. Use the Distributive Property to factor out the GCF.
4. Use the Distributive Property to verify that the factorization is correct.
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 5.4
Page 2 of 3
Example:
10 x 2 y 2  15 xy 3  25 x3 y 4  5 xy 2  2 x  5 xy 2  3 y  5 xy 2  5 x 2 y 2
 5 xy 2   2 x  3 y  5 x 2 y 2 
Practice:
1. 7 z 2  14 z 
2. 6 y 3  14 y 2  10 y 
3. 2m4 n 2  8m3n 4  6m2 n5 
4. 5 y 2  10 y 
5. 3a 3  6a 2  12a 
6. 4a  a  3  3  a  3 
7.
 w  2 w  5   2w  1 w  5 
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 5.4
Page 3 of 3
Factoring by grouping:
1. Group terms that have a common factor. You may need to rearrange the terms.
2. In each grouping, factor out the common factor.
3. Factor out the common factor that remains.
4. Check your work.   .
Example:
5 y  5 z  ay  az   5 y  5 z    ay  az 
 5 y  z   a  y  z 
  5  a  y  z 
Practice:
1. 5 x  5 y  bx  by 
2. w3  3w2  4w  12 
3. 6 z 2  2 z  9 z  3 
4. 2 x 2  x  10 x  5 
5. 6 x3  5  15 x 2  2 x 
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.