Beyond FOIL Alternate Methods for Multiplying and Factoring Polynomials
FOIL Method Distributive Method Box Method Vertical Method Multiplying Polynomials
Distributive Method STEP 1: Rewrite the problem
Distributive Method STEP 2: Distribute
Distributive Method STEP 3: Combine Like Terms
Multiplying Polynomials (5x – 6)(3x + 8) WATCH THOSE SIGNS!!!
Rewrite the problem
Distribute
Combine Like Terms
Binomial x Trinomial Multiplying Polynomials
Rewrite the problem
Distribute
Combine Like Terms
(3x + 2)(5x + 4) Multiplying Polynomials BOX Method
STEP 1: Draw the BOX
Draw the Box 2x2 for a Binomial x Binomial
BOX Method STEP 2: Place terms on outside
BOX Method STEP 3: Multiply: Find the area of each box.
BOX Method STEP 3: Combine Like Terms
BOX Method LET’S SEE THAT AGAIN!
What about a binomial x trinomial?
Vertical Method
How do you multiply without a calculator?
What if we tried it this way?
Can we do that again?
MULTIPLYING POLYNOMIALS (3x + 2)(5x + 4)
VERTICAL Method STEP 1: Rewrite the Problem
VERTICAL Method
STEP 2: MULTIPLY
VERTICAL Method STEP 3: Combine Like Terms
VERTICAL Method
WHAT IF IT’S A TRINOMIAL x A BINOMIAL?
VERTICAL Method STEP 1: Rewrite the Problem
VERTICAL Method
STEP 2: MULTIPLY
VERTICAL Method STEP 3: Combine Like Terms
A SHORTCUT IS NOT A SHORTCUT IF IT IS THE ONLY WAY YOU KNOW.
FIRST FOIL METHOD F
OUTER FOIL METHOD O
INNER FOIL METHOD I
LAST FOIL METHOD L
Kinda
By Grouping GCF Trinomials Factoring Polynomials
Factor Pairs 24 1 · 24 2 · 12 3 · 8 4 · · 40 2 · 20 4 · 10 5 · · 84 2 · 42 3 · 28 4 · 21 6 · 14 7 · 12
Greatest Common Factor 63 1 · 63 3 · 21 7 · · 84 2 · 42 3 · 28 4 · 21 6 · 14 7 · 12
( ) Factor by Grouping 15x xy + 35xz + 28yz 3x ( 5x ) + 7z ( 5x ) + 4y (5x + 4y)(3x+ 7z)
( ) Factor by Grouping 24ac – 9ad – 32bc + 12bd NEGATIVECHANGE -
( ) Factor by Grouping 24ac – 9ad – 32bc + 12bd 3a ( 8c ) - 4b ( 8c ) - 3d (8c – 3d)(3a- 4b) -
Factoring Trinomials without a leading coefficient x 2 + 8x + 15
Factor Start Here Ask Yourself: What are the factor pairs of 15, 1 · 15 3 · 5
x 2 + 8x + 15 Factor Start Here Ask Yourself: What are the factor pairs of 15, 1 · 15 3 · 5 whose sum 1+= 16 3+= 8 is 8?
x 2 + 8x + 15 Factor = 16 3+= 8 x( ) x35++ What signs would make a + 8?
x 2 + 5x - 24 Factor Start Here Ask Yourself: What are the factor pairs of 24, 1 · 24 2 · 12 3 · 8 4 · 6
x 2 + 5x - 24 Factor Start Here Ask Yourself: What are the factor pairs of 24, whose difference is 5? 1 · 24 2 · 12 3 · 8 4 · = 23 = 10 = 5 = 2
( ) x 2 + 5x - 24 Factor = 23 = 10 = 5 = 2 x( )x38-+ What signs would make a + 5?
x 2 – 8x Factor Start Here Ask Yourself: What are the factor pairs of 105, 1 · · 35 5 · 21 7 · 15
· 105 x 2 - 8x Factor Start Here Ask Yourself: What are the factor pairs of 24, whose difference is 8? 1 · 35 5 · 21 7 · 15 =104 = 32 = 16 = 8
x 2 - 8x Factor =104 = 32 = 16 = 8 ( )x x715+- What signs would make a - 8?
Factoring Trinomials with a leading coefficient 6x x + 10
Factor 1 st Step Multiply Leading Coefficient and Constant
Multiply 6x x x 2 nd Step Factor Pairs of 60
Factor Pairs 6x x · 60 2 · 30 3 · 20 4 · 15 5 · 12 6 · 10 3 rd Step Whose sum Is 19. =61 =32 =23 =19
Rewrite 6x x · 60 2 · 30 3 · 20 4 · 15 5 · 12 6 · 10 4 th Step Rewrite the Polynomial =61 =32 =23 =19
Rewrite 6x x · 60 2 · 30 3 · 20 4 · 15 5 · 12 6 · 10 First Term =61 =32 =23 =19 6x 2 Factor Pair 4x15x Last Term + 10 Choose Signs ++
Rewrite 6x x · 60 2 · 30 3 · 20 4 · 15 5 · 12 6 · 10 5 th Step Factor by Grouping =61 =32 =23 =19 6x 2 4x15x+ 10++
( ) 2x 6x 2 ( ) Grouping 6x x · 60 2 · 30 3 · 20 4 · 15 5 · 12 6 · 10 =61 =32 =23 =19 4x15x x ( ) 3x ( ) 3x + 2 (3x + 2)(2x+5)
Factoring With the BOX x 2 – 10x + 16
x 2 -10x + 16 Factor Start Here Ask Yourself: What are the factor pairs of 16, 1 · 16 2 · 8 4 · 4
2 · 1 · x x + 16 Factor Start Here Ask Yourself: What are the factor pairs of 24, whose sum is 10? · 44+ = 17 = 10 = 8
x x + 16 BOX = 17 = 10 = 8 Place terms inside the box x2x2 2x 8x 16
x x + 16 BOX = 17 = 10 = 8 Find the GCF of the columns and rows x2x2 2x 8x 16 x2 x 8
x2x2 2x 8x 16 x2 x 8 (x + 2)(x + 8)
Thank You!! Todd Rackowitz Independence High School