 3x
a) 
 y
4



3
b) 3 24  81
c) 3x  4 y  13x  4 y  1
3
3
Section P.5
Factoring is the process of writing a polynomial as the
product of two or more polynomials.
Prime polynomials cannot be factored using integer
coefficients.
Factor completely means keep factoring until
everything is prime







Greatest common factor
Difference of two perfect squares
Perfect-square trinomials
Factoring x2 + bx + c (big X)
Factoring ax2 + bx + c (big X)
Factor by grouping – use with 4 terms
Sum and difference of perfect cubes



Find the greatest
common factor
(GCF) of all terms.
Divide each term by
the greatest
common factor.
Write the GCF
outside
parenthesis, with
the rest of the
divided terms
added together
inside


3a2 – 12a
3a is the GCF
2
3a
 a,
3a
 12a
 4
3a
3aa  4
a. 18x3 + 27x2
b. x2(x + 3) + 5(x + 3)





Works with an even number of terms.
Split the terms into two groups.
Factor each group separately using GCF.
If factor by grouping is possible, the part
inside parentheses of each group will be
the same.
Treat the parentheses as common factors
to finish factoring.
x  5 x  3 x  15
3
2
x  6 x  2 x  12
3
2

Look for integers r and s such that:
◦ r×s=c
c
◦ r+s=b
r
s
b
x  bx  c  x  r x  s 
2
x  7 x  12
2
x  5x  6
2
x  5x  6
2

Look for integers r and s such that:
◦ r × s = ac
ac
◦ r+s=b
r
s
b


Divide r and s by a, then reduce fractions
In your factors, any remaining denominator
gets moved in front of the x
5 x  13 x  6
2
6 x  11x  3
2
Let A and B be real numbers, variables, or
algebraic expressions,
1. A2 + 2AB + B2 = (A + B)2
2. A2 – 2AB + B2 = (A – B)2

Factor: 16x2 – 56x + 49

Factor: x2 + 14x + 49
a  b  a  ba  b
2
2
x  9   x  3 x  3
2
4 x  1  2 x  12 x  1
2
25 x  49 y  5 x  7 y 5 x  7 y 
2
2

A3  B3   A  B  A2  AB  B 2
Type

Example
A3 + B3 = (A + B)(A2 – AB + B2) x3 + 8 = x3 + 23
= (x + 2)( x2 – x·2 + 22)
= (x + 2)( x2 – 2x + 4)
A3 – B3 = (A – B)(A2 + AB + B2) 64x3 – 125 = (4x)3 – 53
= (4x – 5)((4x)2 + (4x)(5) + 52)
= (4x – 5)(16x2 + 20x + 25)
125x  8 y
3
3
27 x  1000
3
1.
2.
If there is a common factor, factor out the
GCF.
Determine the number of terms in the
polynomial and try factoring as follows:
a)
If there are two terms, can the binomial be factored
by one of the special forms including difference of
two squares, sum of two cubes, or difference of
two cubes?
b) If there are three terms, is the trinomial a perfect
square trinomial? If the trinomial is not a perfect
square trinomial, try factoring using the big X.
c) If there are four or more terms, try factoring by
grouping.
3.
Check to see if any factors with more than
one term in the factored polynomial can be
factored further. If so, factor completely.
Factor out GCF
2
Count number of terms
Check for:
•Difference of perfect squares
•Sum of perfect cubes
•Difference of perfect cubes
3
4
Factor by
Grouping
1. Check for perfect
square trinomial
2. Use big X factoring
Check each factor to see if it can be factored further

Page 53 #1-75 Odd
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

In Exercises 1-10,
factor out the
greatest common
factor.
In Exercises 11-16,
factor by grouping.
In Exercises 17-30,
factor each
trinomial, or state
that the trinomial is
prime.
1) 18 x  27
3) 3 x 2  6 x
5) 9 x  18 x  27 x
7) x x  5  3 x  5
4
3
2
9) x 2  x  3  12 x  3
11) x 3  2 x 2  5 x  10
13) x 3  x 2  2 x  2
15) 3 x 3  2 x 2  6 x  4
17) x 2  5 x  6
19) x 2  2 x  15
21) x  8 x  15
2

In Exercises 17-30,
factor each trinomial, or
state that the trinomial
is prime.
23) 3 x  x  2
2
25) 3 x  25 x  28
2
27) 6 x  11x  4
2
29) 4 x  16 x  15
2

In Exercises 31-40,
factor the difference of
two squares.
31) x  100
2
33) 36 x  49
2
35) 9 x  25 y
2
37) x  16
4
39) 16 x  81
4
2


In Exercises 41-48, factor
any perfect square
trinomials, or state that
the polynomial is prime.
In Exercises 49-56, factor
using the formula for the
sum or difference of two
cubes.
41) x  2 x  1
2
43) x  14 x  49
2
45) 4 x 2  4 x  1
47) 9 x 2  6 x  1
49) x 3  27
51) x 3  64
53) 8 x  1
3
55) 64 x 3  27

In Exercises 57-84, factor
completely, or state that
the polynomial is prime.
57) 3 x 3  3 x
59) 4 x 2  4 x  24
61) 2 x 4  162