Techniques of Factoring
Common Factors
Difference of Two Squares
XBOX Pattern
Factoring Polynomials
This process is basically the REVERSE
of the distributive property.
distributive property
( x  2)(x  5) 
x  3x 10
factoring
2
Factoring Polynomials
In factoring you start with a polynomial
(2 or more terms) and you want to rewrite it
as a product (or as a single term)
Three terms
x  3x 10  ( x  2)(x  5)
2
One term
§ 3. 1
Factoring
The Greatest Common Factor
Factors
Factors (either numbers or polynomials)
When an integer is written as a product of
integers, each of the integers in the product is a
factor of the original number.
When a polynomial is written as a product of
polynomials, each of the polynomials in the
product is a factor of the original polynomial.
Factoring – writing a polynomial as a product of
polynomials.
Greatest Common Factor
Greatest common factor – largest quantity that is a
factor of all the integers or polynomials involved.
Finding the GCF of a List of Integers or Terms
1) Prime factor the numbers.
2) Identify common prime factors.
3) Take the product of all common prime factors.
• If there are no common prime factors, GCF is 1.
Greatest Common Factor
Example
Find the GCF of each list of numbers.
1) 12 and 8
12 = 2 · 2 · 3
8=2·2·2
So the GCF is 2 · 2 = 4.
2) 7 and 20
7=1·7
20 = 2 · 2 · 5
There are no common prime factors so the
GCF is 1.
Greatest Common Factor
Example
Find the GCF of each list of numbers.
1) 6, 8 and 46
6=2·3
8=2·2·2
46 = 2 · 23
So the GCF is 2.
2) 144, 256 and 300
144 = 2 · 2 · 2 · 3 · 3
256 = 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2
300 = 2 · 2 · 3 · 5 · 5
So the GCF is 2 · 2 = 4.
Greatest Common Factor
Example
Find the GCF of each list of terms.
1) x3 and x7
x3 = x · x · x
x7 = x · x · x · x · x · x · x
So the GCF is x · x · x = x3
2) 6x5 and 4x3
6x5 = 2 · 3 · x · x · x
4x3 = 2 · 2 · x · x · x
So the GCF is 2 · x · x · x = 2x3
Greatest Common Factor
Example
Find the GCF of the following list of terms.
a3b2, a2b5 and a4b7
a3b2 = a · a · a · b · b
a2b5 = a · a · b · b · b · b · b
a4b7 = a · a · a · a · b · b · b · b · b · b · b
So the GCF is a · a · b · b = a2b2
Notice that the GCF of terms containing variables will use the
smallest exponent found amongst the individual terms for each
variable.
Factoring Polynomials
The first step in factoring a polynomial is to
find the GCF of all its terms.
Then we write the polynomial as a product by
factoring out the GCF from all the terms.
The remaining factors in each term will form a
polynomial.
Techniques of Factoring
Polynomials
1. Greatest Common Factor (GCF).
The GCF for a polynomial is the largest
monomial that divides each term of the
polynomial.
Factor out the GCF:
4y  2y
3
2
Factoring Polynomials - GCF
4y  2y
3
2
Write the two terms in the
form of prime factors…
22y y y 2 y y
2yy ( 2 y
They have in common 2yy
1)
 2 y (2 y 1)
2
This process is basically the reverse of the distributive property.
Check the work….
2 y (2 y  1)  4y  2y
2
3
2
Factoring Polynomials - GCF
3 terms
Factor the GCF:
4ab  12a b c  8ab c 
3
2
3 2
2
4 a b ( b - 3a c
One term
4 2
+
2
2b c
2
)
Factoring Polynomials - GCF
EXAMPLE:
5x(2 x  4)  3(2 x  4) 
(2 x  4) ( 5x - 3 )
Practice
Factor the following polynomial.
12 x  20 x  3  4  x  x  4  5  x  x  x  x
2
4
 4  x  x (3  5  x  x )
 4 x (3  5 x )
2
2
Practice
Factor the following polynomial.
15 x y  3 x y  3  5  x  y  3  x  y
3
5
2
4
3
5
 3  x  y (5  x  y  1)
2
4
 3 x 2 y 4 (5 xy  1)
2
4
Factoring out the GCF
Practice
Factor out the GCF in each of the following
polynomials.
1) 6x3 – 9x2 + 12x =
3 · x · 2 · x2 – 3 · x · 3 · x + 3 · x · 4 =
3x(2x2 – 3x + 4)
2) 14x3y + 7x2y – 7xy =
7 · x · y · 2 · x2 + 7 · x · y · x – 7 · x · y · 1 =
7xy(2x2 + x – 1)
Factoring out the GCF
Practice
Factor out the GCF in each of the following
polynomials.
1) 6(x + 2) – y(x + 2) =
6 · (x + 2) – y · (x + 2) =
(x + 2)(6 – y)
2) xy(y + 1) – (y + 1) =
xy · (y + 1) – 1 · (y + 1) =
(y + 1)(xy – 1)
§ 3.2
Difference of
Two Squares
Difference of Two Squares
(a + b)(a – b) = a2– ab + ab – b2 = a2 – b2
FORMULA:
a2 – b2 = (a + b)(a – b)
The difference of two bases being squared,
factors as the product of the sum and difference
of the bases that are being squared.
Difference of Two Squares
a2 – b2 = (a + b)(a – b)
A binomial is the difference of two square if
1.both terms are squares and
2.the signs of the terms are different.
9x2 – 25y2
– c4 + d4
Factoring the difference of two squares
2
a
Factor
Difference
of two squares
–
= (a + b)(a – b)
2
b
x2 – 4y2
(x) 2
2
(2y)
(x – 2y)(x + 2y)
Factor
Difference
Of two squares
(
r
4
r2
 25
16
r
 
4
2
– 5)(
2
(5)
r
4
+ 5)
Note: You can use FOIL method to verify that the factorization for the polynomial is accurate.
Difference of two squares
( x  y) ( x  2)
2
2
 ( x  y  ( x  2))(x  y  ( x  2))
 (2 x  y  2)( y  2)
Difference of two squares
y  16
4
 ( y 2 ) 2  ( 4) 2
 ( y 2  4)( y 2  4)
 ( y  2)( y  2)( y 2  4)