MATH JOURNAL ENTRY• SOLVE THIS PROBLEM • (6x2 + 4x - 9) + ( 12x2 + 9x - 13) • TELL WHETHER YOU PERFER TO

GROUP TERMS OR USE THE COLUMN METHOD TO ADD OR SUBTRACT POLYNOMIALS. EXPLAIN WHICH ONE YOU PERFER.

ObjectiveThe student will be able to:

Multiply two polynomials using the FOIL method, Box method and the

distributive property.

AHSGE I-3

There are three techniques you can use for multiplying

polynomials.The best part about it is that they are all

the same! Huh? Whaddaya mean?It’s all about how you write it…Here they

are!1)Distributive Property2)FOIL3)Box Method

Sit back, relax (but make sure to write this down), and I’ll show ya!

1) Multiply. (3x + 2)(6x + 5)

Using the distributive property, multiply 3x(6x + 5) + 2(6x + 5).

18x2 + 15x + 12x + 10Combine like terms.18x2 + 27x + 10

A shortcut of the distributive property is called the FOIL

method.

The FOIL method is ONLY used when you multiply 2 binomials.

It is an acronym and tells you which terms to multiply.

2) Use the FOIL method to multiply the following binomials:

(y + 6)(y + 9).

(y + 6)(y + 9). F tells you to multiply the

FIRST terms of each binomial.

y2

(y + 6)(y + 9). O tells you to multiply the

OUTER terms of each binomial.

y2 + 9y

(y + 6)(y + 9). (y + 6)(y + 9). II tells you to multiply the tells you to multiply the

INNERINNER terms of each binomial. terms of each binomial.

yy22 + 9y + 9y ++ 6y6y

(y + 6)(y + 9). L tells you to multiply the

LAST terms of each binomial.y2 + 6y + 9y + 54

Combine like terms.y2 + 15y + 54

Remember, FOIL reminds you to multiply the:

First terms

Outer terms

Inner terms

Last terms

YOU CAN ALSO USE THIS METHOD TO LABEL IT.

F L F L• ( 8x + 4 ) ( 2x - 3) • O I I O• F *F= 8x*2x F = 16x2 • O*O= 8x * -3 O = -24x I *I=4 *2x I = 8x • L*L= 4*-3 L= -12

TALK –N- TURN• TALK AND TURN TO YOUR PARTNER

AND WORK OUT THE FOLLOWING PROBLEMS.

• DISCUSS WITH YOUR PARTNER WHAT YOU ARE HAVING TROUBLE WITH THE MOST.

• WRITE YOUR ANSWER USING THE PADDLES.

EXAMPLE # 3

( x + 8 ) ( x – 7)

THE ANSWER

x2+ 8x -7x -56

x2 +1x -56

EXAMPLE # 4• Watch Your signs• F L F L

( x - 8) ( x - 9 ) O I I O

THE ANSWER

x2 -9x -8x +72

x2 – 17x +72

THE FABULOUS 7

TICKET OUT THE DOOR• EXPLAIN WHAT YOU HAVE

LEARNED TODAY. • TELL WHAT WAS DIFFICULT ABOUT

THIS LESSON.• WHICH METHOD DO YOU PERFER?

The third method is the Box Method. This method works for every

problem!Here’s how you do it. Multiply (3x – 5)(5x + 2)Draw a box. Write a

polynomial on the top and side of a box. It does not

matter which goes where.This will be modeled in the

next problem along with FOIL.

3x -5

5x

+2

3) Multiply (3x - 5)(5x + 2)

First terms:Outer terms:Inner terms:Last terms:

Combine like terms.

15x2 - 19x – 10

3x -5

5x

+2

15x2

+6x

-25x

-10

You have 3 techniques. Pick the one you like the best!

15x2

+6x-25x-10

4) Multiply (7p - 2)(3p - 4)

First terms:Outer terms:Inner terms:Last terms: Combine like

terms.

21p2 – 34p + 8

7p -2

3p

-4

21p2

-28p

-6p

+8

21p2

-28p-6p

+8

4) Multiply (7p - 2)(3p - 4)

First terms:Outer terms:Inner terms:Last terms: Combine like

terms.

21p2 – 34p + 8

7p -2

3p

-4

21p2

-28p

-6p

+8

21p2

-28p-6p

+8

Multiply (y + 4)(y – 3)1. y2 + y – 12

2. y2 – y – 123. y2 + 7y – 124. y2 – 7y – 125. y2 + y + 126. y2 – y + 127. y2 + 7y + 128. y2 – 7y + 12

Multiply (2a – 3b)(2a + 4b)

1. 4a2 + 14ab – 12b2 2. 4a2 – 14ab – 12b2

3. 4a2 + 8ab – 6ba – 12b2

4. 4a2 + 2ab – 12b2

5. 4a2 – 2ab – 12b2

4th Method to Use• Quita Method• (8v – 2) ( 3v +4)• 8v (3v+4) -2 (3v +4)• Notice! Both binomials are the same• 24v2 + 32v • -6v -8• Combine the middle terms• 24v2 +26v -8•

5th Method to use• Arithmetic Method• (11w -8)(w+3)• Set it up like addition problem. Don’t Add! 11w – 8 w + 3 +33w -2411w2 -8w 11w2 +25w -24Watch your signs

Watch For Signs• You Try! • (2z +1) (10z+4) • (8y -10) (5y - 3)• ( 4p +5) (2p – 9)• (5n - 9 )(5n +5)• Answers• 20z2 +8z +10z+4 = 20z2 +18z+4• 40y2 -24y -50y +30 = 40y2 - 74y +30• 8p2 -36p +10p -45 = 8p2 -26p -45• 25n2 +25n-45n +45 = 25n2 -15n +45

5) 5) Multiply (2x - 5)(xMultiply (2x - 5)(x22 - 5x + 4) - 5x + 4)You cannot use FOIL because they You cannot use FOIL because they are not BOTH binomials. You must are not BOTH binomials. You must

use the distributive property.use the distributive property.

2x2x(x(x22 - 5x + 4) - 5x + 4) - 5- 5(x(x22 - 5x + 4) - 5x + 4)

2x2x33 - 10x - 10x22 + 8x - 5x + 8x - 5x22 + 25x - 20 + 25x - 20

Group and combine like terms.Group and combine like terms.

2x2x33 - 10x - 10x22 - 5x - 5x22 + 8x + 25x - 20 + 8x + 25x - 20

2x2x33 - 15x - 15x22 + 33x - 20 + 33x - 20

x2 -5x +4

2x

-5

5) Multiply (2x - 5)(x5) Multiply (2x - 5)(x22 - 5x + 4) - 5x + 4) You cannot use FOIL because they are not You cannot use FOIL because they are not

BOTH binomials. You must use the BOTH binomials. You must use the distributive property or box method.distributive property or box method.

2x3

-5x2

-10x2

+25x

+8x

-20

Almost done!Go to

the next slide!

x2 -5x +4

2x

-5

5) Multiply (2x - 5)(x5) Multiply (2x - 5)(x22 - 5x + 4) - 5x + 4) Combine like terms!Combine like terms!

2x3

-5x2

-10x2

+25x

+8x

-20

2x3 – 15x2 + 33x - 20

Multiply (2p + 1)(p2 – 3p + 4)

1. 2p3 + 2p3 + p + 4

2. y2 – y – 123. y2 + 7y – 124. y2 – 7y – 12

HOW TO USE ALL FIVE METHODS

• 1. Distributive property(5b+6)(6b +7)

• 2.FOIL( 2c+6) (10c+2)• 3. BOX(2z+8)(z+8)• 4. QUITA (11w+8) (w+3)• 5. Arithmetic(9f+5)(9f +7)

1) Multiply. (3x + 2)(6x + 5)

Using the distributive property, multiply 3x(6x + 5) + 2(6x + 5).

18x2 + 15x + 12x + 10Combine like terms.18x2 + 27x + 10

A shortcut of the distributive property is called the FOIL

method.

YOU CAN ALSO USE THIS METHOD TO LABEL IT.

F L F L• ( 8x + 4 ) ( 2x - 3) • O I I O• F *F= 8x*2x F = 16x2 • O*O= 8x * -3 O = -24x Combine

the I *I=4 *2x I = 8x -16x • L*L= 4*-3 L= -12 Middle• THE ANSWER 16x2 – 16x -12

4) Multiply (7p - 2)(3p - 4)

First terms:Outer terms:Inner terms:Last terms: Combine like

terms.

21p2 – 34p + 8

7p -2

3p

-4

21p2

-28p

-6p

+8

21p2

-28p-6p

+8

4th Method to Use• Quita Method• (8v – 2) ( 3v +4)• 8v (3v+4) -2 (3v +4)• Notice! Both binomials are the same• 24v2 + 32v • -6v -8• Combine the middle terms• 24v2 +26v -8•

5th Method to use• Arithmetic Method• (11w -8)(w+3)• Set it up like addition problem. Don’t Add! 11w – 8 w + 3 +33w -2411w2 -8w 11w2 +25w -24Watch your signs

5) 5) Multiply (2x - 5)(xMultiply (2x - 5)(x22 - 5x + 4) - 5x + 4)You cannot use FOIL because they You cannot use FOIL because they are not BOTH binomials. You must are not BOTH binomials. You must

use the distributive property.use the distributive property.

2x2x(x(x22 - 5x + 4) - 5x + 4) - 5- 5(x(x22 - 5x + 4) - 5x + 4)

2x2x33 - 10x - 10x22 + 8x - 5x + 8x - 5x22 + 25x - 20 + 25x - 20

Group and combine like terms.Group and combine like terms.

2x2x33 - 10x - 10x22 - 5x - 5x22 + 8x + 25x - 20 + 8x + 25x - 20

2x2x33 - 15x - 15x22 + 33x - 20 + 33x - 20

x2 -5x +4

2x

-5

5) Multiply (2x - 5)(x5) Multiply (2x - 5)(x22 - 5x + 4) - 5x + 4) You cannot use FOIL because they are not You cannot use FOIL because they are not

BOTH binomials. You must use the BOTH binomials. You must use the distributive property or box method.distributive property or box method.

2x3

-5x2

-10x2

+25x

+8x

-20

Almost done!Go to

the next slide!

x2 -5x +4

2x

-5

5) Multiply (2x - 5)(x5) Multiply (2x - 5)(x22 - 5x + 4) - 5x + 4) Combine like terms!Combine like terms!

2x3

-5x2

-10x2

+25x

+8x

-20

2x3 – 15x2 + 33x - 20