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MATH JOURNAL ENTRY SOLVE THIS PROBLEM (6x 2 + 4x - 9) + ( 12x 2 + 9x - 13) TELL WHETHER YOU PERFER TO GROUP TERMS OR USE THE COLUMN METHOD TO ADD OR SUBTRACT POLYNOMIALS. EXPLAIN WHICH ONE YOU PERFER.

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Objective The student will be able to: Multiply two polynomials using the FOIL method, Box method and the distributive property. AHSGE I-3

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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 Property 2)FOIL 3)Box Method Sit back, relax (but make sure to write this down), and I’ll show ya!

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1) Multiply. (3x + 2)(6x + 5) Using the distributive property, multiply 3x(6x + 5) + 2(6x + 5). 18x 2 + 15x + 12x + 10 Combine like terms. 18x 2 + 27x + 10 A shortcut of the distributive property is called the FOIL method.

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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).

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(y + 6)(y + 9). F tells you to multiply the FIRST terms of each binomial. y2y2

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(y + 6)(y + 9). O tells you to multiply the OUTER terms of each binomial. y 2 + 9y

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(y + 6)(y + 9). I tells you to multiply the INNER terms of each binomial. y 2 + 9y + 6y

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(y + 6)(y + 9). L tells you to multiply the LAST terms of each binomial. y 2 + 6y + 9y + 54 Combine like terms. y 2 + 15y + 54

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Remember, FOIL reminds you to multiply the: F irst terms O uter terms I nner terms L ast terms

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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 = 16x 2 O*O= 8x * -3 O = -24x I *I=4 *2x I = 8x L*L= 4*-3 L= -12

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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.

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EXAMPLE # 3 ( x + 8 ) ( x – 7)

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THE ANSWER x 2 + 8x -7x -56 x 2 +1x -56

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EXAMPLE # 4 Watch Your signs F L F L ( x - 8) ( x - 9 ) O I I O

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THE ANSWER x 2 -9x -8x +72 x 2 – 17x +72

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THE FABULOUS 7

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TICKET OUT THE DOOR EXPLAIN WHAT YOU HAVE LEARNED TODAY. TELL WHAT WAS DIFFICULT ABOUT THIS LESSON. WHICH METHOD DO YOU PERFER?

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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

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3) Multiply (3x - 5)(5x + 2) First terms: Outer terms: Inner terms: Last terms: Combine like terms. 15x 2 - 19x – 10 3x-5 5x +2 15x 2 +6x -25x -10 You have 3 techniques. Pick the one you like the best! 15x 2 +6x -25x -10

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4 ) Multiply (7p - 2)(3p - 4) First terms: Outer terms: Inner terms: Last terms: Combine like terms. 21p 2 – 34p + 8 7p-2 3p -4 21p 2 -28p -6p +8 21p 2 -28p -6p +8

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4 ) Multiply (7p - 2)(3p - 4) First terms: Outer terms: Inner terms: Last terms: Combine like terms. 21p 2 – 34p + 8 7p-2 3p -4 21p 2 -28p -6p +8 21p 2 -28p -6p +8

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Multiply (y + 4)(y – 3) 1.y 2 + y – 12 2.y 2 – y – 12 3.y 2 + 7y – 12 4.y 2 – 7y – 12 5.y 2 + y + 12 6.y 2 – y + 12 7.y 2 + 7y + 12 8.y 2 – 7y + 12

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Multiply (2a – 3b)(2a + 4b) 1.4a 2 + 14ab – 12b 2 2.4a 2 – 14ab – 12b 2 3.4a 2 + 8ab – 6ba – 12b 2 4.4a 2 + 2ab – 12b 2 5.4a 2 – 2ab – 12b 2

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

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5 th Method to use Arithmetic Method (11w -8)(w+3) Set it up like addition problem. Don’t Add! 11w – 8 w + 3 +33w -24 11w 2 -8w 11w 2 +25w -24 Watch your signs

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

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5) Multiply (2x - 5)(x 2 - 5x + 4) You cannot use FOIL because they are not BOTH binomials. You must use the distributive property. 2x(x 2 - 5x + 4) - 5(x 2 - 5x + 4) 2x 3 - 10x 2 + 8x - 5x 2 + 25x - 20 Group and combine like terms. 2x 3 - 10x 2 - 5x 2 + 8x + 25x - 20 2x 3 - 15x 2 + 33x - 20

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x2x2 -5x+4 2x -5 5) Multiply (2x - 5)(x 2 - 5x + 4) You cannot use FOIL because they are not BOTH binomials. You must use the distributive property or box method. 2x 3 -5x 2 -10x 2 +25x +8x -20 Almost done! Go to the next slide!

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x2x2 -5x+4 2x -5 5) Multiply (2x - 5)(x 2 - 5x + 4) Combine like terms! 2x 3 -5x 2 -10x 2 +25x +8x -20 2x 3 – 15x 2 + 33x - 20

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Multiply (2p + 1)(p 2 – 3p + 4) 1.2p 3 + 2p 3 + p + 4 2.y 2 – y – 12 3.y 2 + 7y – 12 4.y 2 – 7y – 12

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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)

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1) Multiply. (3x + 2)(6x + 5) Using the distributive property, multiply 3x(6x + 5) + 2(6x + 5). 18x 2 + 15x + 12x + 10 Combine like terms. 18x 2 + 27x + 10 A shortcut of the distributive property is called the FOIL method.

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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 = 16x 2 O*O= 8x * -3 O = -24x Combine the I *I=4 *2x I = 8x -16x L*L= 4*-3 L= -12 Middle THE ANSWER 16x 2 – 16x -12

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4 ) Multiply (7p - 2)(3p - 4) First terms: Outer terms: Inner terms: Last terms: Combine like terms. 21p 2 – 34p + 8 7p-2 3p -4 21p 2 -28p -6p +8 21p 2 -28p -6p +8

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

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5 th Method to use Arithmetic Method (11w -8)(w+3) Set it up like addition problem. Don’t Add! 11w – 8 w + 3 +33w -24 11w 2 -8w 11w 2 +25w -24 Watch your signs

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5) Multiply (2x - 5)(x 2 - 5x + 4) You cannot use FOIL because they are not BOTH binomials. You must use the distributive property. 2x(x 2 - 5x + 4) - 5(x 2 - 5x + 4) 2x 3 - 10x 2 + 8x - 5x 2 + 25x - 20 Group and combine like terms. 2x 3 - 10x 2 - 5x 2 + 8x + 25x - 20 2x 3 - 15x 2 + 33x - 20

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x2x2 -5x+4 2x -5 5) Multiply (2x - 5)(x 2 - 5x + 4) You cannot use FOIL because they are not BOTH binomials. You must use the distributive property or box method. 2x 3 -5x 2 -10x 2 +25x +8x -20 Almost done! Go to the next slide!

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x2x2 -5x+4 2x -5 5) Multiply (2x - 5)(x 2 - 5x + 4) Combine like terms! 2x 3 -5x 2 -10x 2 +25x +8x -20 2x 3 – 15x 2 + 33x - 20

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