Beyond FOIL Alternate Methods for Multiplying and Factoring Polynomials.

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Presentation transcript:

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