lesson 5-4 & 5-5: factoring
DESCRIPTION
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. If there is a GCF factor it out!!!!!! Ex: 2x + 8 2(x + 4). Trinomial (3 terms). Polynomial (4 terms). - PowerPoint PPT PresentationTRANSCRIPT
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
factor flow chartRemember: Number of exponent tells you number of Factors/ Solutions/ Roots/ Interceptsx1 = 1 factorx2 = 2 factorsx3 = 3 factorsx4 = 4 factors and so on…..
If there is a GCF factor it out!!!!!!Ex: 2x + 8
2(x + 4)
Binomial(2 terms)
Trinomial(3 terms)
Polynomial (4 terms)
Is it a difference of squares or
cubes ?A2- B2 or A3±B3
ex: 4x2 – 25 or x3 - 64
Done
No
Difference of Squares (DS)A2- B2 = (A +B )(A – B)Ex: 4x2 – 25 = (2x + 5)(2x - 5)
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)
Is it a Perfect Square Trinomial?
A2 ± 2AB + B2
ex: 4x2-20x +25
(2x-5)2
PST A2 +2AB+B2 = (A + B)2
Or A2 -2AB+B2 = (A - B)2
Ex: 4x2 –20x +25 = (2x - 5)2
yes
No
Find
Write out factors
If a=1 If a≠1
Rewrite as four terms
Factor by: GroupingOr Undo foil( )( ) or box
ac b
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 What is the GCF?
Pull out GCF (divide both terms)
3uv2
3uv2(2uv - 7)
Factoring 4-term
Make Sure Polynomial is in descending order!!!!!!!!
3 Methods
A) Reverse FOIL
F O I L
x2 + 5x + 4x + 20
( )( )
REMEMBER: ALWAYS FACTOR A GCF 1st IF YOU CAN
Find GCF of first two terms- fill first spot
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!!x x + 5+ 4
• Foil Box
F
x2
O
+ 5xI
+ 4xL
+ 20
x2 + 5x + 4x + 20
x + 5
x
+ 4
( x + 5)(x + 4)
B) Factor by grouping
x2 + 5x + 4x + 20
x( x + 5 )
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
+ 4(x + 5)
(x + 5) (x - 4)
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) =
So to factor we are unFOILing!!
2x2 - 6x + 4x – 122x2 - 2x - 12
Steps for General Trinomial Factoring1) 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
Example 1: x2 + 7x + 12
F O I L
( )( )
2) ac
1*12
b
7
x2 + 4x + 3x + 12
1*12 13
2*6 8
3*4 7
x x + 4+ 3
12
1) no GCF
TRY
Example 2 Factor x2 – 5x – 24
Example 3 Factor x2 – 12x + 27
EX 4) Harder One
6x2 – 5x – 4
-8x
+ 1
- 4+ 3x6x2
F O I L
( ) ( )2x
GCF of first 2
3x - 4
-24
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
Ex 2: 9x2 – 16y2 Difference of squares so
(x - 4)2
(3x + 4y)(3x – 4y)
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: )168(64 2 xx
Marker Board pg 222-223
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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?
a
+b
a3
b3
a2 + b2
a2b
-a2b
-ab
-ab2
ab2
(a3 + b3)= ( a+b)(a2-ab+b2) Is the remaining trinomial factorable?
Ex 2: Factor 27x3-8y3
or (3x)3 _ (2y)3
27x3
-8y3
3x
-2y
9x2 +4y2
-18x2y
18x2y +12xy2
-12xy2
+ 6xy
27x3-8y3=(3x-2y)(9x2+6xy+4y2)
A3 – B3 = (A-B)(A2 + AB+ B2)
Ex 3: Factor x3 + 64
Formulas
A3 – B3 = (A-B)(A2 + AB+ B2)
A3 + B3 = (A+B)(A2 - AB+ B2)
Factor : 125x3 +1
Marker Board pg 227
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