Lesson 5-4 & 5-5: Factoring
Objectives:
Students will:
•Factor using GCF
•Identify & factor square trinomials
•Identify & factor difference of two squares
Day 1
• Trinomials
Remember:
Number of exponent tells you number of
Factors/ Solutions/ Roots/ Intercepts
x1 = 1 factor
x2 = 2 factors
x3 = 3 factors
x4 = 4 factors and so on…..
factor flow chart
If there is a GCF factor it out!!!!!!
Ex: 2x + 8
2(x + 4)
Trinomial
(3 terms)
Binomial
(2 terms)
Is it a difference of
squares or cubes
?
A2- B2 or A3±B3
ex: 4x2 – 25 or x3
- 64
yes
Polynomial
(4 terms)
Is it a Perfect
Square Trinomial?
A2 ± 2AB + B2
ex: 4x2-20x +25
(2x-5)2
No
No
PST
Find
A2 +2AB+B2 = (A +
B)2
Or A2 -2AB+B2 = (A - B)2
Ex: 4x2 –20x +25 = (2x 5)2
Done
Difference of Squares
(DS)
= (A +B )(A – B)
Ex: 4x2 – 25 = (2x + 5)(2x - 5)
A2-
B2
Repeat with (ax-b) if
possible
Difference (or sum) of Cubes
(A3 – B3) = (A - B)(A2 +AB + B2)
Or (A3 + B3) = (A + B)(A2 - AB + B2)
(then factor trinomial if possible)
Ex: x3 – 64 = (x – 4)(x2 + 4x + 16)
ac
If a=1
Write out
factors
b
If a≠1
Rewrite as
four terms
Factor by:
Grouping
Or
Undo foil
(
)(
box
) or
Factoring
The reverse of multiplying
2x(x+3) = 2x2 + 6x
So:
2x2 + 6x =
Look for GCF of all terms → numbers & variables
► Reverse distribute it out → DIVISION
Example 1 Factor 6u2v3 – 21uv2
Pull out GCF (divide both terms)
3uv2(2uv - 7)
What is the GCF? 3uv2
Factoring 4-term
Make Sure Polynomial is in descending order!!!!!!!!
3 Methods
Find GCF of first two terms- fill first spot
A) Reverse FOIL
F
O
I
L
x2 + 5x + 4x + 20
( x + 4 )( x + 5 )
Find what makes up ( F) and fill in first spot in other
factor already have x so need another x
Move to outside (O) already have x so need + 5
Move to inside (I) already have x so need + 4
Check last (L) 4x5 =20 so done!!
REMEMBER: ALWAYS FACTOR A GCF 1st IF YOU CAN
• Foil Box
x2 + 5x + 4x + 20
( x + 5)(x + 4)
x +5
x
F
O
x2
I
+4
+ 4x
+ 5x
L
+ 20
B) Factor by grouping
x2 +
5x + 4x + 20
x( x + 5 ) + 4(x + 5)
(x + 5) (x - 4)
Find GCF of first two terms- and factor out
Find GCF of second two terms- and factor out
What is in parenthesis should match –so factor it
out
Write what is left as other factor
It’s the same either
method!!
I like the FOIL method. What do you think????
ax2 + bx + c
– A General Trinomial
Where does middle term come from?
(x + 2)(x + 3) = x2 + 3x + 2x + 6
(2x + 4)(x – 3) = 2x2 - 6x + 4x – 12
2x2 - 2x - 12
So to factor we are unFOILing!!
Steps for General Trinomial Factoring
1) Factor out GCF (always first step)
2) Find product ac that add to b table (to find O and I)
3) Write middle term as combo of factors ( 4 terms)
4)Unfoil or by grouping
12
Example 1:
x2 + 7x + 12
1) no GCF
2)
ac
1*12
b
7
1*12
13
2*6
8
3*4
7
x2 + 4x + 3x + 12
F
O
I
L
( x + 3 )( x + 4
)
TRY
Example 2 Factor
Example 3 Factor
x2 – 5x – 24
x2 – 12x + 27
EX 4) Harder One
-24
6x2 – 5x – 4
6x2
-8x
F
O
+ 3x - 4
I
( 2x + 1 ) ( 3x
GCF of first 2
L
- 4 )
Factor:
-7a + 6a2 -10
Factor: 56 + x – x2
Assignment (day 1)
• 5-5/227/ 22-72 e
Day 2
Factoring Perfect Squares, Difference of Square,
Look back at the forms for each of these from Lesson 5-3
Factor the following:
Ex 1: x2 – 8x + 16
Perfect Square Trinomial so
(x - 4)2
Ex 2: 9x2 – 16y2
(3x + 4y)(3x – 4y)
Difference of squares so
Ex 3:
Factor 8x2 – 8y2
Don’t forget
GCF!
Trick: Ex 4:
Combo perfect square trinomial and difference of squares
x2 – 2xy + y2 – 25
(x-y)2 - 25
((x-y) + 5)((x-y) – 5)
Apply PST
Now apply DS
Ex 5:
Factor:
64  ( x 2  8x  16)
Marker Board pg 222-223
•
•
•
•
21
33
41
51
ASSIGNMENT
• 5-4/222-223/18-62e, 86-92 e
Day 3
• Sum or Difference of cubes
Review Cubing Binomials
• (a+b)3= (a+b)(a2 +2ab+b2)
a3 +3a2b+3ab2+b3
(similarly for (a-b)3)
Example 1:
(a3 + b3)
Notice all the middle terms cancelled out like DS.
What were the terms that cancelled?
What are the factors?
a2
a
a3
+b
a2b
-ab
-a2b
-ab2
(a3 + b3)= ( a+b)(a2-ab+b2)
+ b2
ab2
b3
Is the remaining
trinomial
factorable?
Ex 2: Factor 27x3-8y3
or (3x)3 _ (2y)3
9x2
+ 6xy
+4y2
+12xy2
3x
27x3
18x2y
-2y
-18x2y
-12xy2
27x3-8y3=(3x-2y)(9x2+6xy+4y2)
A3 – B3 = (A-B)(A2 + AB+ B2)
3
-8y
Ex 3: Factor x3 + 64
Formulas
3
A –
3
B
=
2
(A-B)(A
+ AB+
2
B)
A3 + B3 = (A+B)(A2 - AB+ B2)
Factor : 125x3 +1
Marker Board pg 227
• 1
• 13
• 19