Factoring by Grouping
Factoring:
Some rules governing factoring in elementary algebra:
1. The numbers (coefficients) must be real and rational.
2. The powers must be whole numbers.
3. Prime factors include:
a) Linear polynomial, such as (x + 3), (x + y + z)
b) The sum of an even degree term (2 or higher) with any real number, such as:
(x^{2 }+ 4), (x^{2 }y^{2 } + 9), (2x^{2 } + 25), (x^{2
} + 3), etc.
c) The sum of two evenpowered terms, such as: (x^{4} + y^{4}), (x^{6} + y^{8}), (x^{2} + y^{6})
d) Certain quadratic trinomials which cannot be expressed as the product of two linear
binomials with rational numbers. There is a special test for this which will be discussed
later.
Factor Quadratic Polynomials:
WHAT TO DO: 
HOW TO DO IT: 
1. First, examine the polynomial to see if it has a
common factor(s) in each term. Integers as
common factors are left in composite form and
letters are left in power form. Note the remaining
factor is prime. 
a) 3x^{2} + 12
3(x^{2} + 4) prime factors
b) 5x^{3} + 7x^{2}
x^{2}(5x + 7) prime factors 
Factor out that common factor(s) and see what is
left. If the remaining polynomial is prime*, leave
the factors â€œas isâ€.

c) 3x^{3} − 21x^{2} + 27x
3x(x^{2} − 7x + 9) prime factors 
If the inner polynomial is factorable, set it aside
to examine separately and come back to complete. 
d) 4x^{2} − 12x − 8x + 24
4(x^{2} − 3x − 2x + 6) 
2. Consider the inner polynomial: x2 − 3x − 2x + 6
grouped 2Ã—2 by underlining the first two and last two.
Factor common factor from each group.
Bring down middle sign.
Complete factoring 

Now, go back to the original problem and write
the answer in completely factored form. 
d) 4x^{2} − 20x + 24
4(x^{2} − 5x + 6)
4(x − 3)(x − 2) 
