Algebra 3
Warm-Up 2.2
List the factors of 36
1,
2,
3,
4,
6,
36
18
12
9
Algebra 3
Lesson 2.2
Objective:
SSBAT factor a polynomial by factoring out the GCF
and using Difference of Squares.
Standards: M11.D.2.2.2
Monomial
An expression with 1 term
5x2
-3mn3k
8
Binomial
An expression with 2 terms
4x – 2
15x3 + 8y
Trinomial
An expression with 3 terms
8x5 – 5w3 + 2
16 – 3x + 5m4
Factors
 The numbers used in a multiplication problem
 5 x 3 = 15, 5 and 3 are the factors of 15
List the Factors of 24
 1, 2, 3, 4, 6, 8, 12, 24
Greatest Common Factor (GCF)
The biggest number that is a factor of all of the
numbers in a set.
Find the GCF of 18 and 45
 Factors of 18:
1, 2, 3, 6, 9, 18
 Factors of 45:
1, 3, 5, 9, 15, 45
 GCF of 18 and 45 is 9
Finding the GCF of expressions (variables)
Example: 6x4y and 10x2
1. Find the GCF of the Coefficients (numbers in front)
2. Find the GCF of each variable piece by
 Look at only one set of like variables at a time
 If the variable does not appear in all of the terms
do not use it in the GCF
 If the variable appears in every term use the one
with the smaller exponent in the GCF
Find the GCF of each.
1.
25x2y4 and 10x3y
 GCF: 5x2y
2.
21mn2k and 10m2n2
 mn2
3.
12x4y,
9x5y2, 21x7yw3
 GCF: 3x4y
4.
m5n3k2 and mnk
 GCF: mnk
Factoring
 Rewriting an expression as a multiplication problem
 2 · 5 is the factored form of 10
 3(x + 4) is the factored form of 3x + 12
Factoring Out the GCF
1. Find the GCF of all of the terms in the polynomial
2. Write the GCF outside of the parentheses
3. Divide each term of the polynomial by the GCF
and write this expression inside the parentheses
Examples: Factor out the GCF of each.
1.
2w3 + 10w
The GCF is 2w
= 2w(w2 + 5)
Examples: Factor out the GCF of each.
2.
18n3 + 9n2 – 24n
The GCF is 3n
=
3n(6n2 + 3n – 8)
Examples: Factor out the GCF of each.
3.
–20x6 + 12x3 – 4x
The GCF is 4x  When the lead coefficient is Negative factor out
the negative as well
=
–4x(5x5 – 3x2 + 1)
Examples: Factor out the GCF of each.
4.
15mn3 – 9m2n4 + 18m3n5
The GCF is 3mn3
= 3mn3(5 – 3mn + 6m2n2)
Perfect Square
 A number that you can take the Square Root of
 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121,…
 Perfect Square expressions are x2, x4, x6, …
Difference of Squares
 An expression of the a2 – b2
 Perfect Square – Perfect Square
Examples:
x2 – 4
m2 – 25
9x2 – 1
*** x2 + 4 is NOT a difference of squares because of the PLUS ***
Factoring a Difference of Squares
a2 – b2 = (a + b)(a – b)
 Take the square root of each term
 Add the 2 square roots in one ( )
 Subtract the 2 square roots in another ( )
Factor each Difference of Squares
1.
x2 – 64
= ( x + 8 )( x – 8 )
2.
81 – x2
=( 9 + x )( 9 – x)
Factor each Difference of Squares
3.
4x2 – 25
= ( 2x + 5 )(2x – 5 )
4.
w 2 – y2
= ( w + y )(w – y )
Factor each Difference of Squares
5.
49x2 – 1
= ( 7x + 1 )( 7x – 1 )
6.
x2 – 130
Can’t Do – It is NOT a Difference of Squares  130 is not a
perfect square
7.
x2 + 9
Can’t Do – It is NOT a Difference of Squares  It’s Plus not
Minus
8.
w6 – 196
= (w3 + 14)(w3 – 14)
9.
100x22 – y16
= (10x11 – y8)(10x11 + y8)
On Your Own.
1.
Factor out the GCF.
8x3 – 20x5 + 2x2
= 2x2(4x – 10x3 + 1)
2.
Factor the Difference of Squares. 36x2 – 49
= (6x + 7)(6x – 7)
3.
Factor out the GCF. 12m3n5 – 24mn4 – 30m6n
= 6mn(2m2n4 – 4n3 – 5m5)
4.
Factor the Difference of Squares. 81 – m12
= (9 + m6)(9 – m6)
Homework
Worksheet 2.2