lesson 5-4 & 5-5: factoring objectives: students will: factor using gcf identify & factor...

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Lesson 5-4 & 5-5: Factoring Objectives: Students will: •Factor using GCF •Identify & factor square trinomials •Identify & factor difference of two squares

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Page 1: Lesson 5-4 & 5-5: Factoring Objectives: Students will: Factor using GCF Identify & factor square trinomials Identify & factor difference of two squares

Lesson 5-4 & 5-5: Factoring

Objectives:Students will:•Factor using GCF•Identify & factor square trinomials•Identify & factor difference of two squares

Page 2: 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

Page 3: Lesson 5-4 & 5-5: Factoring Objectives: Students will: Factor using GCF Identify & factor square trinomials Identify & factor difference of two squares

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

Page 4: Lesson 5-4 & 5-5: Factoring Objectives: Students will: Factor using GCF Identify & factor square trinomials Identify & factor difference of two squares

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)

Page 5: Lesson 5-4 & 5-5: Factoring Objectives: Students will: Factor using GCF Identify & factor square trinomials Identify & factor difference of two squares

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

Page 6: Lesson 5-4 & 5-5: Factoring Objectives: Students will: Factor using GCF Identify & factor square trinomials Identify & factor difference of two squares

• Foil Box

F

x2

O

+ 5xI

+ 4xL

+ 20

x2 + 5x + 4x + 20

x + 5

x

+ 4

( x + 5)(x + 4)

Page 7: Lesson 5-4 & 5-5: Factoring Objectives: Students will: Factor using GCF Identify & factor square trinomials Identify & factor difference of two squares

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????

Page 8: Lesson 5-4 & 5-5: Factoring Objectives: Students will: Factor using GCF Identify & factor square trinomials Identify & factor difference of two squares

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

Page 9: Lesson 5-4 & 5-5: Factoring Objectives: Students will: Factor using GCF Identify & factor square trinomials Identify & factor difference of two squares

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

Page 10: Lesson 5-4 & 5-5: Factoring Objectives: Students will: Factor using GCF Identify & factor square trinomials Identify & factor difference of two squares

TRY

Example 2 Factor x2 – 5x – 24

Example 3 Factor x2 – 12x + 27

Page 11: Lesson 5-4 & 5-5: Factoring Objectives: Students will: Factor using GCF Identify & factor square trinomials Identify & factor difference of two squares

EX 4) Harder One

6x2 – 5x – 4

-8x

+ 1

- 4+ 3x6x2

F O I L

( ) ( )2x

GCF of first 2

3x - 4

-24

Page 12: Lesson 5-4 & 5-5: Factoring Objectives: Students will: Factor using GCF Identify & factor square trinomials Identify & factor difference of two squares

Factor: -7a + 6a2 -10

Page 13: Lesson 5-4 & 5-5: Factoring Objectives: Students will: Factor using GCF Identify & factor square trinomials Identify & factor difference of two squares

Factor: 56 + x – x2

Page 14: Lesson 5-4 & 5-5: Factoring Objectives: Students will: Factor using GCF Identify & factor square trinomials Identify & factor difference of two squares

Assignment (day 1)

• 5-5/227/ 22-72 e

Page 15: Lesson 5-4 & 5-5: Factoring Objectives: Students will: Factor using GCF Identify & factor square trinomials Identify & factor difference of two squares

Day 2

Page 16: Lesson 5-4 & 5-5: Factoring Objectives: Students will: Factor using GCF Identify & factor square trinomials Identify & factor difference of two squares

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)

Page 17: Lesson 5-4 & 5-5: Factoring Objectives: Students will: Factor using GCF Identify & factor square trinomials Identify & factor difference of two squares

Ex 3:

Factor 8x2 – 8y2 Don’t forget

GCF!

Page 18: Lesson 5-4 & 5-5: Factoring Objectives: Students will: Factor using GCF Identify & factor square trinomials Identify & factor difference of two squares

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

Page 19: Lesson 5-4 & 5-5: Factoring Objectives: Students will: Factor using GCF Identify & factor square trinomials Identify & factor difference of two squares

Ex 5:

Factor: )168(64 2 xx

Page 20: Lesson 5-4 & 5-5: Factoring Objectives: Students will: Factor using GCF Identify & factor square trinomials Identify & factor difference of two squares

Marker Board pg 222-223

• 21

• 33

• 41

• 51

Page 21: Lesson 5-4 & 5-5: Factoring Objectives: Students will: Factor using GCF Identify & factor square trinomials Identify & factor difference of two squares

ASSIGNMENT

• 5-4/222-223/18-62e, 86-92 e

Page 22: Lesson 5-4 & 5-5: Factoring Objectives: Students will: Factor using GCF Identify & factor square trinomials Identify & factor difference of two squares

Day 3

• Sum or Difference of cubes

Page 23: Lesson 5-4 & 5-5: Factoring Objectives: Students will: Factor using GCF Identify & factor square trinomials Identify & factor difference of two squares

Review Cubing Binomials

• (a+b)3= (a+b)(a2 +2ab+b2)

a3 +3a2b+3ab2+b3

(similarly for (a-b)3)

Page 24: Lesson 5-4 & 5-5: Factoring Objectives: Students will: Factor using GCF Identify & factor square trinomials Identify & factor difference of two squares

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?

Page 25: Lesson 5-4 & 5-5: Factoring Objectives: Students will: Factor using GCF Identify & factor square trinomials Identify & factor difference of two squares

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)

Page 26: Lesson 5-4 & 5-5: Factoring Objectives: Students will: Factor using GCF Identify & factor square trinomials Identify & factor difference of two squares

Ex 3: Factor x3 + 64

Page 27: Lesson 5-4 & 5-5: Factoring Objectives: Students will: Factor using GCF Identify & factor square trinomials Identify & factor difference of two squares

Formulas

A3 – B3 = (A-B)(A2 + AB+ B2)

A3 + B3 = (A+B)(A2 - AB+ B2)

Page 28: Lesson 5-4 & 5-5: Factoring Objectives: Students will: Factor using GCF Identify & factor square trinomials Identify & factor difference of two squares

Factor : 125x3 +1

Page 29: Lesson 5-4 & 5-5: Factoring Objectives: Students will: Factor using GCF Identify & factor square trinomials Identify & factor difference of two squares

Marker Board pg 227

• 1

• 13

• 19

Page 30: Lesson 5-4 & 5-5: Factoring Objectives: Students will: Factor using GCF Identify & factor square trinomials Identify & factor difference of two squares