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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
Multiplying Polynomials
(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
Multiplying Polynomials
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.
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
Start HereAsk Yourself:
What are the factor pairs of 15,
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
Start HereAsk Yourself:
What are the factor pairs of 24,
1 · 242 · 123 · 84 · 6
x2 + 5x - 24
Factor
Start HereAsk Yourself:
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
Start HereAsk Yourself:
What are the factor pairs of 105,
1 · 1053 · 355 · 217 · 15
7-5-3 ·
105x2 - 8x - 105
Factor
Start HereAsk Yourself:
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
Start HereAsk Yourself:
What are the factor pairs of 16,
1 · 162 · 84 · 4
2 ·1 ·
x2 - 10x + 16
Factor
Start HereAsk Yourself:
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)
