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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.