beyond foil alternate methods for multiplying and factoring polynomials

Beyond FOILAlternate Methods for

Multiplying and

Factoring Polynomials

FOIL Method

Distributive MethodBox Method

Vertical Method

Multiplying Polynomials

Distributive Method

)45)(23( xx

STEP 1:

Rewrite the problem

Rewrite the problem

)45)(23( xx

x3 )45( x 2 )45( x

Distributive Method

x3 )45( x 2 )45( x

STEP 2:Distribute

Distribute

x3 )45( x 2 )45( x

215x x12 x10 8

Distributive Method215x x12 x10 8

STEP 3:Combine Like

Terms

Combine Like Terms215x x12 x10 8

215x x22 8


(5x – 6)(3x + 8)

WATCH THOSE SIGNS!!!

)83)(65( xxRewrite the problem

x5 )83( x 6 )83( x

x5 )83( x 6 )83( x

Distribute

215x x40 x18 48

215x x40 x18 48

215x x22 48

Combine Like Terms

Binomial x Trinomial

)145)(23( 2 xxx


Rewrite the problem

)145)(23( 2 xxx

)145(2)145(3 22 xxxxx

Distribute)145(2)145(3 22 xxxxx

281031215 223 xxxxx

281031215 223 xxxxx

Combine Like Terms

25215 23 xxx

(3x + 2)(5x + 4)

Multiplying PolynomialsBOX Method

BOX Method

)45)(23( xx

STEP 1: Draw the

BOX

Draw the Box)45)(23( xx

2x2 for a Binomial x Binomial

BOX Method

)45)(23( xx

STEP 2: Place terms on outside

)45)(23( xx

x3 2x5

4

BOX Method

)45)(23( xx

STEP 3: Multiply: Find the area of each box.

)45)(23( xx

x3 2x5

4

215x xx 35 25 x x10

x34 x12 24 8

BOX Method

)45)(23( xx

STEP 3: Combine Like

Terms

x3 2x5

4

215x x10

x12 8

215x x22 8

BOX Method

LET’S SEE THAT

AGAIN!

BOX Method

)92)(74( xx

BOX Method)92)(74( xx

x4 7x2

9

)92)(74( xx

x4 7x2

9

28x xx 24 72 x x14

x49 x36 79 63

)92)(74( xx

x4 7x2

9

28x x14

x36 63

28x x50 63

BOX Method

)452)(34( 2 xxx

What about a binomial x trinomial?

)452)(34( 2 xxxx4 3

22x

x5

4

38x 26x220x x15

x16 12

x4 322x

x5

4

38x 26x220x x15

x16 1238x 214x x 12

Vertical Method

How do you multiply without a calculator?

3458

34582

3

270

2

0172791

What if we tried it this way?

3458

3458

430850322402001500

3224020015002791

Can we do that again?

6379

6379

360970275402104200

2754021042007794

MULTIPLYING POLYNOMIALS

(3x + 2)(5x + 4)

VERTICAL Method

)45)(23( xx

STEP 1: Rewrite the

Problem

VERTICAL Method

)23( x )45( x

23 x45 x

VERTICAL Method

STEP 2: MULTIPLY

23 x45 x

23 x45 x8x12

x10215x

VERTICAL Method

)45)(23( xx

STEP 3: Combine

Like Terms

8x12x10215x

23 x45 x

8x22215x

VERTICAL Method

)57( x )83( x

57 x83 x

57 x83 x40x56

x15221x

40x41221x

40x56x15221x

57 x83 x

WHAT IF IT’S A TRINOMIAL x

A BINOMIAL?

VERTICAL Method

)34)(235( 2 xxx

STEP 1: Rewrite the

Problem

)235( 2 xx )34( xVERTICAL Method

235 2 xx34 x

VERTICAL Method

STEP 2: MULTIPLY

235 2 xx34 x

235 2 xx34 x6x9215x

x8212x320x

VERTICAL Method

STEP 3: Combine Like

Terms

)34)(235( 2 xxx

235 2 xx34 x6x9215x

x8212x320x6x17227x320x

A SHORTCUT IS NOT A SHORTCUT IF IT IS THE ONLY WAY YOU KNOW.

FIRST

FOIL METHOD

)45)(23( xx

F

215x

OUTER

FOIL METHOD

)45)(23( xx

O

215x x12

INNER

FOIL METHOD

)45)(23( xx

I

215x x12 x10

LAST

FOIL METHOD

)45)(23( xx

L

215x x12 x10 8

FOIL METHOD

)45)(23( xx215x x12 x10 8

82215 2 xx

FOIL METHOD

)752)(34( 2 xxx

Kinda

38x 220x x28 26x x15 21

38x 214x x13 21

By GroupingGCF

Trinomials

Factoring Polynomials

Factor Pairs

241 · 242 · 123 · 84 · 6

401 · 402 · 204 · 105 · 8

841 · 842 · 423 · 284 · 216 · 147 · 12

Greatest Common Factor

631 · 633 · 217 · 9

841 · 842 · 423 · 284 · 216 · 147 · 12

( ) ( )

Factor by Grouping

15x2 + 12xy + 35xz + 28yz3x 3x

3x(5x )+

7z 7z

7z(5x )+ 4y + 4y

(5x + 4y)(3x + 7z)

( ) ( )

Factor by Grouping

24ac – 9ad – 32bc + 12bd

NEGATIVE

CHANGE

-

( ) ( )

Factor by Grouping

24ac – 9ad – 32bc + 12bd3a 3a

3a(8c )-

4b 4b

4b(8c )- 3d - 3d

(8c – 3d)(3a - 4b )

-

FactoringTrinomials without

a leading coefficient

x2 + 8x + 15

x2 + 8x + 15

Factor

Start HereAsk Yourself:

What are the factor pairs of 15,

1 · 153 · 5

x2 + 8x + 15

Factor



1 · 153 · 5

whose sum

1+

= 163+

= 8

is 8?

x2 + 8x + 15

Factor

155

1+

= 163+

= 8

x( )( )x3 5+ +

What signs wouldmake a + 8?

x2 + 5x - 24

Factor



1 · 242 · 123 · 84 · 6

x2 + 5x - 24

Factor


What are the factor pairs of 24,whose difference

1-2-

is 5?

1 · 242 · 123 · 84 · 63-4-

= 23= 10

= 5= 2

( )4-3-

x2 + 5x - 24

Factor1-2-

241286

= 23= 10

= 5= 2

x( ) x3 8- +What signs wouldmake a + 5?

x2 – 8x - 105

Factor



1 · 1053 · 355 · 217 · 15

7-5-3 ·

105x2 - 8x - 105

Factor


What are the factor pairs of 24,whose difference

1-3-

is 8?

1 ·35

5 · 217 · 15

=104= 32= 16= 8

3-

7-5-

105x2 - 8x - 105

Factor1-

352115

=104= 32= 16= 8

( )x( ) x7 15+ -What signs wouldmake a - 8?

FactoringTrinomials with

a leading coefficient

6x2 + 19x + 10

Factor6x2 + 19x + 10

1st StepMultiply Leading Coefficient and

Constant

Multiply6x2 + 19x + 10 60

x

2nd StepFactor Pairs

of 60

Factor Pairs6x2 + 19x + 10 60

1 · 602 · 303 · 204 · 155 · 126 · 10

3rd StepWhose sum

Is 19.

=61=32=23=19

Rewrite6x2 + 19x + 10 60

1 · 602 · 303 · 204 · 155 · 126 · 10

4th StepRewrite thePolynomial

=61=32=23=19

Rewrite6x2 + 19x + 10 60

1 · 602 · 303 · 204 · 155 · 126 · 10

First Term

=61=32=23=19

6x2

Factor Pair

4x 15x

Last Term

+ 10

Choose Signs

+ +

Rewrite6x2 + 19x + 10 60

1 · 602 · 303 · 204 · 155 · 126 · 10

5th StepFactor byGrouping

=61=32=23=19

6x2 4x 15x + 10+ +

( )2x6x2( )

Grouping6x2 + 19x + 10 60

1 · 602 · 303 · 204 · 155 · 126 · 10

=61=32=23=19

4x 15x +10+ +2x

2x( )3x + 2

5 5

+5

( )3x + 2

(3x + 2)

(2x

+5)

FactoringWith the BOX

x2 – 10x + 16

x2 -10x + 16

Factor



1 · 162 · 84 · 4

2 ·1 ·

x2 - 10x + 16

Factor


What are the factor pairs of 24,whose sum

1+2+

is 10?

168

3 · 44+

= 17= 10= 8

1+2+4+

x2 - 10x + 16

BOX1684

= 17= 10= 8

Place terms

inside the box

x2 2x

8x 16

1+2+4+

x2 - 10x + 16BOX

1684

= 17= 10= 8

Find the GCF of the columns and rows

x2 2x

8x 16

x 2

x

8

x2 2x

8x 16

x 2

x

8

(x + 2)

(x + 8)

Thank You!!

Todd RackowitzIndependence High

[email protected]

beyond foil alternate methods for multiplying and factoring polynomials

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