Dividing Polynomials

Home , Polynomial long division

CorrectionKey=NL-D;CA-D DO NOT EDIT--Changes mustbemadethrough “File info” LESSON esn 6 .5 Lesson 321 Performance Task.Performance functionsboth of time. Then preview Lesson the and how number the of teams and attendance the are View Engage the online. section photo the Discuss dividend whentheremainder is0. synthetic division. The ofthe divisorisafactor long divisionor, for adivisoroftheform x-a, Possible answer: You candividepolynomialsusing Essential Question: dividing polynomials. Work insmallgroups to complete acompare for andcontrast chart Objective Language Practices Mathematical A-APR.A.1, A-APR.B.3 longdivision,….Also the formq(x)+r)/b),…usinginspection, Rewrite simplerational expressions indifferent forms;write a(x)/b)in The student isexpectedto: Common Core Math Standards PolynomialsDividing the dividend? of you whenthedivisorisafactor know ways to dividepolynomials, andhow do PERFORMANCE TASK PREVIEW: LESSON ENGAGE COMMON COMMON CORE CORE MP.2 Reasoning A-APR.D.6 6 . 5 6.5 What are some CorrectionKey=NL-B;CA-B DO NOT EDIT--Changes mustbemadethrough "File info" A2_MNLESE385894_U3M06L5 321

© Houghton Mifflin Harcourt Publishing Company You up can set an array of numbers that captures sequence the of multiplications and   p that evaluating it involves an alternating sequence of additions and multiplications. For instance, nested the form of Polynomials written can be nested insomething form. called Apolynomial innested form is written insuch away Explore Name iiig Polynomials Dividing Essential Question: What are someways to dividepolynomials, andhow doyou when know 6.5 oue 6 Module The first two steps are to bring down leading the number, multiply 4,then by value the you Given Given the value of x,addingthe value 3,multiplying byx, adding 2,multiplying by x,and adding 1. ( x ) Add 3and –8. Given Given

= 4 Multiply. Add. Multiply. Multiply. 12 Add. 10 Add. Rewrite additions top find needed are evaluating at, -2. solid down arrows indicate adding. arrow indicates bringing down, diagonal the arrows represent multiplication by –2, x 3

+ 3 p p ( ( x x x p ) 2 )

( + 2x = 4 = 4 -5 x -24 ) asp -8 x x · · the divisor is a factor ofthedividend? the divisorisafactor -2 · 3 + 1is p 3 Evaluating aPolynomial Function ( ( Using Synthetic Substitution + 3 + 3 -2 -2 + 3= + 1= + 2=12 ( · x x 4 x ) ) ) 2

2 = = 10 = = ( + 2x + 2x x ) ( -5 -8 -23 -24 x

DO NOT EDIT--Changes must be made

CorrectionKey=NL-B;CA-B a = ( ) - x - . Using array this top find 1, find p + 1,find x 1, find p + 1,find ( 2 2 ( 4x

Name Dividing Polynomials 6. 5

Essential Question: What are some ways to x

Explore COMMON

Polynomials can be written in something cal CORE that evaluating it involves an alternati A2_MNLESE385894_U3M06L5 321 p the value of x, adding 3, multiplying by x, addin (  ( x

)

= 4 A-APR.D.6 Rewrite simple rational expressions in d Given 4x … using inspection, long division, … Also

You can set up an array o

Rewrite + 3 © Houghton Mifflin Harcourt Publishing Company Given x Multiply.

Add. 3 The first two steps are to br Multiply.

Add. 10 + 3 Multiply. 12 Add.

additions needed to find p  p

x arrow indicates bringing dow solid down arrows indicate adding. (

2 x are evaluating at, -2. through "File info" Module 6 + 2x Add 3 and 8. ) the divisor is a factor of the dividend? – p

= 4 ( + 3

x Evaluating a Polynomial Function p ) Using Synthetic Substitution ( as p x + 1 is p x 4 4

-5 4 4 ) -8 -2

+ 3 24 = 4 -

(

· ) + 3 = · x x · -2 x 2 12 -2 + = ( ) 2 4 3 x + 1 = + 2x = ) + 3 =

= = 10 x 8 ng sequence of additions and multiplications. For instan = - x -5 ( x + 2

x ) 2 ( + 1, find p x -24 f numbers that captures the seque ( + 2x -23 ( 4x (

( 4x (

a divide polynomials, and how do you know whe led nested form. A polynomial in nested form is wri

) Class 3 . Using this array to find p + + 3 -2 ing down the leading number, 4,

n, the diagonal arrows represen + 2

A-APR.A.1, A-APR.B.3 -2 + 1, find p

- g 2, multiplying by x, and adding 1. ) )

2 + 2 ( Class + 2 - -2 ifferent forms; write

)

) ) + 1, which you evaluate by starting with 4, m .

1. 4 +

(

4 -2 )

4 )

) 4 ) by using synthetic substitution. )

. by using synthetic substitution. The dashed + 1. -

8 Date

___ a(x) b

- + 1,which you evaluate by starting with 4,multiplying by ( ( - 3 a x 8

) - 321 )

- 5 in the form

nce of multiplications and -

is called synthetic substituti 321 t multiplication by –2, 321 then multiply by the value you

2 3 8 2 5 3 8

______q

(

)

b + r

(

x 1 ) ce, the nested form of ( x n

) tten in such a way

1 ,

The dashed

ultiplying by

and the

Resource

on Locker

. ( a )

Lesson 5 synthetic substitution is called synthetic HARDCOVER

6/27/14 7:54 PM 2 2 edition. hardcover student find thislessoninthe Turn to thesepagesto edition. hardcover student find thislessoninthe Turn to thesepagesto Date 1 1 PAGES 231240 and the and the . Resource Locker esn 5 Lesson 16/10/14 9:35AM

) )

x ) ( 322 to find to divisor ) remainder x ______( )(quotient )(quotient + = quotient

, and you are finding the are , and you ) divisor dividend ______Dividing Polynomials x - a ( . . Point out that polynomial long division division long polynomial that out . Point ) a ( What operation are you doing and what are are what and doing you are operation What usesynthetic when finding you you Students have the option of completing the completing of the option have Students in the book either activity division polynomial You are dividing the polynomial function are p You ? ) a ( EXPLORE EXPLORE EXPLAIN 1 1 EXPLAIN INTEGRATE TECHNOLOGY INTEGRATE QUESTIONING STRATEGIES QUESTIONING AVOID COMMON ERRORS COMMON AVOID or online. or which is equivalent to dividend = (divisor to equivalent is which the divisor. by term each multiply if you + remainder division long arithmetic of example an Showing may division polynomial of example an alongside the connection. make students help substitution on the polynomial function p Evaluating a Polynomial Function Function Polynomial a Evaluating Substitution Synthetic Using Dividing Polynomials Using Long Long Using Dividing Polynomials Division the familiar difficulty relating have may Students to numbers whole for process long-division the using polynomials for the process identifying dividend = (divisor finding for algorithm + remainder this result: leads to value of p value by the quantity by p DO NOT EDIT--Changes must be made through "File info" "File through made be must EDIT--Changes NOT DO CorrectionKey=NL-B;CA-B 6/27/14 7:01 PM

) ) ) x x x 1 ( ( ( 24 23 R D Lesson Lesson 5 - - _____ +

) , where + r, where is a linear a linear is x ) ) 2 ( 10 12 x ( x - a

= Q

( ) ) ) 8 3 5 x x x ( ( - - ( . This last expression expression last . This P D , we would be able to to be able would , we _____ ) ) x x ( ( = Q )

. Using the . Using 4 4

x (

+ R ) . (This can be can used. (This as . Multiplying through through . Multiplying

1 ( 12 divisor 2 __ -2. remainder ______- D  = + )

the last addition. the last x and then carry then and the out remainder ( reach you until steps of this sequence Continue = 23 +

) 0, we can write write can ≠ 0, we 12  277 ___ quotient ) 322 = Q x where x where = ) ) (

x x remainder ( ( 1 quotient ( + ) divisor dividend ______

) divisor (

+ 1 , where D , where 2 ) = is a polynomial whose degree is less than that of D of that than whose less is degree a polynomial is 10

+ 5 ) x ( x + 3x

(

2 x

( were not less than the degree of D of the degree than less not were + 3x

)

2 ← Dividend ← Remainder ← Quotient

x ÷ 8 x 5 3

( ) - - and D 37 36 ) 24 277 + 2 = (divisor) (quotient)

––––

Dividing Polynomials Using Long Division Long Using Dividing Polynomials ― ―

3 x 1 ⟌ + 5 + 5 ( x 4 12

––––––––––––––––– ⟌ to? does correspond this sum what the final After addition, + 3x + 3x

4 4

-23

2 + 1 x x find the quotient and remainder. Write the result in the form in the form the result Write remainder. and find the quotient dividend a check. as addition and multiplication =

a polynomial Given divisor dividend, and division to use long ) + 2 + 2

3 2 + 3x 3

-2 x

2 x - ( 4 x of 0 (if any). of p ( The final sum represents the value of p the value final sum represents The Discussion

2. – by answer the previous Multiply

PROFESSIONAL DEVELOPMENT PROFESSIONAL where the remainder R the remainder where (If the degree of R of the degree (If number. a real r is the remainder Division of polynomials is related to division of whole numbers. Given Given numbers. whole of division to related is polynomials of Division polynomials P Math Background Math can be used to justify the Remainder Theorem. Notice that when D that Notice be Theorem. can used justify the to Remainder Equivalently, P Equivalently, dividing.) continue , the expression becomes P becomes x - a, the expression the form of divisor Example 1

numbers. when dividing would you as way in the same division Write a coefficient with terms including form, in standard Begin the dividend writing by 4 numbers from above, the arithmetic long division leads to leads to division long the arithmetic above, numbers from a means of checking your work.) your checking a means of  Notice that the long division leads to the result the result leads to division the long that Notice 23 Divisor Explain 1 follows. as proceeds division long Recall arithmetic that Reflect 1.   Module Module 6 by the divisor yields the result dividend the result yields the divisor by

A2_MNLESE385894_U3M06L5 322

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CorrectionKey=NL-D;CA-D DO NOT EDIT--Changes must be made through “File info” “File through be made must EDIT--Changes DO NOT CorrectionKey=NL-D;CA-D DO NOT EDIT--Changes mustbemadethrough “File info” other factor isthequotient.other factor remainder, where isthedivisorand onefactor plusthe should bewrittenoffactors astheproduct has degree 0. has adegree lessthanthedegree ofthedivisor, or esn 6 .5 Lesson 323 QUESTIONING STRATEGIES polynomial division problem? What do you answer write as final the for a polynomial division problem? How can you tell ifyou are solving finished a The answer The remainder CorrectionKey=NL-B;CA-B DO NOT EDIT--Changes mustbemadethrough "File info" A2_MNLESE385894_U3M06L5 323

© Houghton Mifflin Harcourt Publishing Company examples of dividing polynomials write journals. intheir they when quotient and remainder inpolynomial form. Encourage students as to these use solved and steps all are explained. The last student summarizes by giving the explains it. continue They to pass problem the until each problem is completely have pass them problem the to another student, writes who next the step and tothem show and explain first step the individing with synthetic division. Then Provide each student with adifferent example of polynomial division, and ask Help groups of students practice dividing polynomials using synthetic division. Small Group Activity COLLABORATIVE LEARNING

Module 6 Module

(

Write dividend the instandard form, including terms of with acoefficient 0.

x Write division the same inthe way as you would dividing when numbers. subtract that value from dividend. the Next, multiply divisor the through by term of the quotient the you just found and with first term the of dividend. the In inorder case, this to get 4 Find value the you to need multiply divisor the by that so first term the matches Since 29x 4 Write answer. final the 4x Check. x

x subtract 4 x term of remaining the dividend is of lower degree than divisor. the Taking difference this dividend, as new the continue fashion inthis until largest the 6

-10

2 2

2 2 2

6 x

4 by

3 3 + 3x + 2x + 3x + 3x 4

+ 2 + 2 + 5 4

+ 5 . This will be the first term the of be quotient. the x. This will x x x - 5 + 1 + 1 + 1 2 2

x x

+ 5is of lower degree than + 3x + 3x + 2x ――――――― - ―――――――― - ――――――― - 3

⟌ ⟌ ⟌ ⟌ + 12 + 0 ( –––––––––––––––––––––– ––––––––––––––––– ––––––––––––––––– ––––––––––––––––– ( (

6 4 4 4 4x 4x 4 -10 -10 4 4x x x x x x x + 5= + 5= + 8

4 3

x - 10

x + 5 + 2 + 2 + 2 + 12 + 12

2 x x 2

from 4 +4xfrom + 2x

2 2

) x = = 4

- - 30x 2

÷ x x x x

- x 3 2 2

x 4x ( (

x x x 2

x + 0 + 3x + 3x + 3x 2

(

29x

x 2 2

x + 4x 3 3

+ 4x + 5 + 8

+ 3x + 3x + 2 + 12 + 5

+ 2x - 10 x + 15 2

+ 5 + 5 + 5

) ) x + 2x

x x

+ 1 + 1

) 2

+ 3x

- 5

+ 4x + 2 ) ) + 8 ( ( ) x 4x 4x (

x + 5

2 x 2

+ 3x 2

- 10 - 10 + 3x

+ 3x 10 x ) ) + 1,stop. 29x + 5. 2

+ 1

+ 29x + 29x 323 - 30x ) ( 4x + 15 + 15 - 10+29x )

= 4 + 15is remainder. the x 3

+ 12 x

2 ,we must multiply + 15 x 2

, so +4x,so

Lesson 5 Lesson 6/27/14 7:01PM 324 Dividing Polynomials DO NOT EDIT--Changes must be made through "File info" info" "File through made be must EDIT--Changes NOT DO CorrectionKey=NL-B;CA-B 6/27/14 7:01 PM

Lesson 5 - 10 + - 2 + 5x - 220 - 121x + 63x

) - 60x

+ 35x 2

- 10

x x 5x ( ) + 30

- 30 2

- 12 + 228 + 1 x + 228 - 30 + 8x + 88x + 6x

- 2 3 3

2 -121x x x

x x - 121x )

3 ) (

63x 15

-14 = 15 =

+ 44 + 44 x 324

- 7x )

+ 12 - 7x

2 x + 8 and then carry then and a check. out x x 6

( 6

) - 2 - 10 ) (

) ) + 8 + 44 + 2x

- 5

3 3

63x - 12 )

) - 5 - 12 x x )

2 - 60x + 8

- 220 x + 228 + 2x

remainder

+ 1

2 44 - 7 + 5 x + 5x + 35x + 2x

+ 3x

x +

4 4 + 2x 2

+ 8x 2

+ 2x x x )

- 30 x 2

x x +

x 2

(

3 x x + 6x + 0 x -30

+ 88x

(

x - 33x

3 ( -121x

= 6 = 6 x 2

x 7x -14 x - ――――――――― 3 -30 + 30 + 0 x

( + 30

+ 12 - 10 44 -

+ 5

3 x x 44 + 8 = quotient

4 ÷ 2

4 x x

( + 8 = x

( ) x x ) 5x 15 6 15 6 6

–––––––––––––––––––– ( (

-7 –––––––––––––––––––––– -7 - ⟌ ⟌ ( - ―――――――――― - ―――――――― ――― + 2x

――― + 2x

- - 12 3

3 + 1 x - 5 x divisor

( Have them show how to to how them show . Have

+ 8x + 5

+ 5 =

+ 2x 4

+ 6x

x 2

Check. x Divide. Write the final answer. Write 6 6 x

15 (

How do you include the terms with coefficients of 0? coefficients with of the terms include you do How You represent the term with 0 as the coefficient, e.g, 0x. e.g, with 0 as the coefficient, the term represent You

3 Reflect

DIFFERENTIATE INSTRUCTION DIFFERENTIATE Have groups of students create create students of groups Have them help to organizers graphic synthetic using polynomials divide division a form into a problem organize have Then shown. the one to similar of each write to them use organizers the results. then interpret the and cells, of each into goes what explain the steps, Graphic Organizers Graphic Your Turn Your

Module 6 3. in the form the result Write remainder. and find the quotient divisionto long Use dividend 2.

A2_MNLESE385894_U3M06L5 324

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CorrectionKey=NL-D;CA-D DO NOT EDIT--Changes must be made through “File info” “File through be made must EDIT--Changes DO NOT CorrectionKey=NL-D;CA-D DO NOT EDIT--Changes mustbemadethrough “File info” Focus onPatterns Synthetic Division variable by decreased 1. of quotient, the starting with power the of the last row, 3;and last the row includes coefficients the rows are remainder the added; is last the digit inthe the formthe of divisor the is Students should notice that divisor the is -2because understand patterns. the For example, arrows and expressions, ifnecessary, to help them patterns insynthetic division. Have students use esn 6 .5 Lesson 325 ( MP.8 Dividing ÷ PRACTICES INTEGRATE MATHEMATICAL EXPLAIN 2 2 x down 2 ( bring 2

x +2

+ 7x9 - 2 Students should quickly that see there are ) may shown be as: 2 2 p ) (

÷ x -2(2) ) ( by x by x +2 - 3 7 4 ) ( 7+(-4) -

x -a = 2x+3 a Using -2(3) ) ,or - ( x - 9 3 6 ( 2 _____ 9+(-6) x +2 x ( 2

-2

+ 7x9

) ) ; the the ; )

CorrectionKey=NL-B;CA-B DO NOT EDIT--Changes mustbemadethrough "File info" A2_MNLESE385894_U3M06L5 325

© Houghton Mifflin Harcourt Publishing Company the divisor.the that equation for a.This procedure remind will students correct to the for sign use divisor syntheticthe division problem. To help students remember value the of afor the havewill to understand how to interpret numerical the results last in the row of synthetic representation of apolynomial division problem. Point out that English the word Help students division understand how synthetic method the Connect Vocabulary LANGUAGE SUPPORT Check. Write result, the using non-remainder the Bring down Then multiply first coefficient. the By inspection,a For reason, this synthetic substitution. division synthetic called is also of left the p  is thatis p Compare long division with synthetic substitution. There are Explain2 two important things to notice. The first 4. oue 6 Module Long Division Example 2

2 x -

-133

9 (

( 9 entries of bottom the row as coefficients. the and add for each column. and and ――――― -

2 7 x (

( ⟌ x 16 + ( 4

–––––––––––––––––– a x -a

means not genuine, means unnatural, artificial, or contrived

3 3

3 3 ) +

a x x - 6x isto equal remainder the p when x ( ―――――― - , use synthetic division to divide by x to division divide synthetic p(x),use Given apolynomial 3

a ―――――――― - in the syntheticin the division format.

- 6 + 4 ) + 10x20 ( ⟌ 3

in the bottom inthe row of synthetic the substitution array give of coefficients the quotient. the addition as acheck. p (x) quotient andthe (nonzero) the remainder. Write result the form inthe 10 10 (

–––––––––––––––––––––––

+ 11 9 9 + 9 ) x x x x x x x ,show how them to form equation the x-a=0,and solve then 2

) 4

10 + 0x - 20x + 0x

―――― -20x40 + + 0 + ) x ――――――― -

――――――――― - 2

20x +10 ÷ (

x x = - 4 ( x Dividing Dividing (

3 x x x - 133 -133 (

) - + 11 -5. Write coefficients the

3 3

3 x

) + 144 + 0 - 133

+ a 50 ÷ )

(quotient)

5 x x x ( x ) x

2 2

-16x

+ 0x - 16x x + 16x

x + 0x + 16

2 )

+ 0x

Synthetic Substitution ) 4 - + 2124 - 2128

( )

2 34010 - 4 + p

x ( 3102050

) ( ) 6 2 40 20 6 ――――――

7 is divided by x by x by ( x a 3 )

) - 6x then carry out multiplication the and ,then carry

- 325 + 9

( 7

a Using Synthetic Division )

3 = = 7 = 7 9x = = 9

-5 - 6x -5 ( a. The second is that numbers the to ( 16x x x x x

3 3 2

+ 5

4 4

- 6x - 35

+ 16 + 9 + + + 2124 ) x x ( ) 7

3 3

x 7 ――――――――― 7

= + 9 x )

―――――― 7 + 11 - 133 2

( 2

- 35 9 - 35x ( -35 x -35 x 0

+ 5 0

x +

x -6

2 x is used as asymbolic is used

. That implies they - 4

- ) x + 169 - 175x

( + 144

7 -6 169 - 133 175 a and obtain x 9 2

- 35x

)

-836 -845 - 836 x + 169x )

- 16x + 16x 9 + 169

+ 845-836 + 2124 - 2128 )

- 836 esn 5 Lesson 16/10/14 9:40AM 326 Dividing Polynomials The bottom row of the synthetic division division of the synthetic row bottom The How can you recognize the quotient and and the quotient recognize you can How division the synthetic when using remainder QUESTIONING STRATEGIES QUESTIONING problem gives the coefficients of the quotient along along quotient of the coefficients the gives problem division problem. of the with the remainder method? DO NOT EDIT--Changes must be made through "File info" info" "File through made be must EDIT--Changes NOT DO CorrectionKey=NL-B;CA-B 6/27/14 7:01 PM

+ 3x - 34 + 41

) 2 9x - 6 + 41 +

+ 17x 2 ) + 9x 9x

2 - + 17

+ 5 - 18x 3

2 x 27x + 7 x 3 is a quadratic binomial. + 3 is a quadratic

x - + 9x

2 - 3 + 5

- 3x 3 2

x + 9 - 3 - 27 x

2 2 3 6x 2 a; and obtain obtain a and ( x x + x (

)

) 2 25x + 5 -

3 1 x _

) + 5 - 4 + 5 + 2 -

+ 3? Explain. 3 3 - 2 - 4 4

+ )

2 x x 3 x x x + 6 x x 6 - 2 + 7 6x (

(

+ 6

= 2 = 2 =

= - 2 - 2x

2 326 - 2x x

2 + 6x x

+ 2 2 + 2

x 3 + 2 x

3 4 - 2 + 7x x (

3 2 ) 4

( x x 2 1 _

)

2 1

) _

- )

+ 2 - 3 1 3

1 2

_ x

4 4 - )

. You may wish to perform a check. perform wish to may . You

( x x ) + x - 2 a 7 5 2 (

x ( x - 2

(

( = 4 = 4 = x 7 =

( ) ÷ ) ÷

+ p

) , use synthetic divide division to x by -6 -3 -3 )

) ÷ )

2 5 + + 2 ) x -3 3 9 9 ( 7 0 + 5 + 2 41

7x 7 + 7 0 6 + 7x -1

-2 -27 - 3x 2 1 2 quotient 1 -1 4 ――――――― _ + 7x

-25 x

- ( 3 x -3

) ――――――― 2 6 3 5 x 9 6 a 2 1 x _

- 3

25x 0 2 2 -2 4 - in the synthetic division format. division a in the synthetic the coefficients and write a. Then a =

- x - 3 4 2

- + 5 4

x x 4 ――――――

( 4 ――――――― 4

1 3 4 3 _

( x

x x 4

( Find Find Find Find

Check. Bring down the first coefficient. Then multiply and add for each column. each for add and the coefficient. first multiply Then down Bring Write theWrite result. -

6 2 2 1 ( _

( ( 2 2 Can you use synthetic division to divide a polynomial by by a polynomial divide to division use synthetic Can you x - must be a linear binomial in the form the divisor No,

) Reflect x

Your Turn Your (

Module 6 6. a polynomialGiven p 5.

B the quotient and the (nonzero) remainder. Write the result in the form in the form the result Write remainder. the (nonzero) and the quotient p

A2_MNLESE385894_U3M06L5 326

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CorrectionKey=NL-D;CA-D DO NOT EDIT--Changes must be made through “File info” “File through be made must EDIT--Changes DO NOT CorrectionKey=NL-D;CA-D DO NOT EDIT--Changes mustbemadethrough “File info” that if ofthepolynomial.a factor The Factor Theorem says by x-a,andtheremainderby p polynomial asthequotient q p esn 6 .5 Lesson 327 Factor Theorem Using theRemainder Theorem and Theorem impliesthat ifapolynomialp QUESTIONING STRATEGIES EXPLAIN 3 ( x ) = ( Factor Theorem related? How are Remainder the Theorem and the ( x -a x -a ) ) is a factor ofp isafactor q ( x ) . ( ( x ( x ) a ) ,you canrewrite the ) timesthisfactor, or The Remainder =0,then ( x ) isdivided ( x -a ) is CorrectionKey=NL-B;CA-B DO NOT EDIT--Changes mustbemadethrough "File info" A2_MNLESE385894_U3M06L5 327

q Write Since remainder the is 0,x Write Since remainder the is , So, So, of are Theorem facts 0.These known as Factor the where where p the the If remainder the p ( ( Example 3 x a ) ) Remainder Theorem Remainder

x = = -1 4 Use synthetic division.

q q p p - q -3 14

( ( p

( ( ( ( x x ( ( ( ――――――――

x x x a x x ) ) x a is afactor of p 1 ) ) p

) ) ) - a -

= = p = = ( ――――――― is quotient the and ris anumber. Substituting a is divided by x = =

x ( x x = Determine whether the given the binomial Determine is p whether afactor of polynomial the x 0 1

) a q

x x

) ) ) x x x x

( (

( = 12 0 -3 so, remaining the find factors of p -5 1 1-5 5 -1

q q 3 4

x x x =

2 3 3

2 ( ( + 3 - 4 )

( 2 x a - 5 - 5 - x +

x ( and factor then it. ) x ) - 4

a ( ,and you when divide p

- + x - 4 4 1 )

0 -4 x x + 3 inp 1 ) -6 45 x x ) 2 3

= by grouping. by 5 r. Since a

) (

2 2 - 4x - 6 ( x

Using theRemainder Theorem andFactor ) x Theorem - - ( ( x x

. 3

( x - x -

-12 x 2

- - 6 - )

x x x ) + - 4x .Conversely, ifx

( 1 2

= , the resulta, the written can be form inthe p - 12; +5 +5

5 x + 4x ) 2 x

) ( 0

- 5 ) ( 0 2

x ( x

x + 4x - 12= - - + 3is afactor. a - ( ( ) + 5; x x 0, this simplifies= 0,this to p a

5 2 ) + 3 + 1 ) q ) + 5=

(

( x x ( ) ) ) (

x + + 1 afactor. Write q - ais afactor of p ) + 2 by x by is p ( ) a

) ) ( . is p 0,then x - a,you get quotient the q (

- 2 ( x x ) + 327 . ) ( 1 x for for ( ) + 3 a ( ) ( x in this equationx inthis gives

x ( = x + ) )

) . = r. This is known as , then you ,then can write p ( 1 x ( ) ) ( x . ( x x - )

= - a ) ( 1 q x ) ( ( x - ( x ) x ) ,which tells you a with aremainder - 5 ) q ( x ) )

+ ( x ) r as as ( . x ) .If esn 5 Lesson 16/10/14 9:42AM 328 You use You Different synthetic division? division? synthetic Dividing Polynomials Alike Synthetic division is a division process that that division is a division process Synthetic When do you use synthetic substitution, and and substitution, use synthetic you do When use whenyou do How do you explain the process of synthetic synthetic of the process explain you do How useful? is it why when and and division, Have students work in pairs to complete a a complete to in pairs work students Have Long Division Synthetic Substitution or Division ELABORATE ELABORATE QUESTIONING STRATEGIES QUESTIONING MATHEMATICAL INTEGRATE PRACTICES SUMMARIZE THE THE SUMMARIZE LESSON uses only the coefficients of a polynomial. If the last of a polynomial. uses only the coefficients sum is 0, then the binomial is a factor of the down not bringing are polynomial; possible errors of add instead to forgetting the first coefficient, include coefficients to subtract, and forgetting 0. are that What are possible sources of error in the error of sources possible are What process? Focus on Communication Focus MP.3 chart like the following, showing similarities and and similarities showing the following, like chart differences: synthetic substitution when you want to find the find the to want substitution when you synthetic a certainfunction of a polynomial for number. value find to want you division when use synthetic You of polynomial division. the quotient DO NOT EDIT--Changes must be made through "File info" info" "File through made be must EDIT--Changes NOT DO CorrectionKey=NL-B;CA-B 28/06/14 4:10 PM

) + 1 Lesson 5 x -

2 x

( ) ) + 4 + 1 is a factor. x x ) ( ( ) 2 + 4 = - 1 is not a factor.

+ 1 x + 8 ) ( x . If it is, find it . If - + 1

)

3 x + 2x x

( x 3

( ( ) x + 1 + 8

x x ( + 2 = 2

x = 2 = 2

) x ( = 2

p ) x ( 328 So, q Since the remainder is 6, x the remainder Since Since the remainder is 0, remainder the Since

) How do you know when the divisor is a factor of the dividend? of a factor is when the divisor know you do How + 4

x 0 8 ) ( . -8

)

- 1 x + 8; 2 0 ( 2 x (

5 1 6 0 0 0 + 2x

+ 5;

3 x 3 1 -2 0 8 - 2x + 8

-8

3 4 0 x x 3 3 ―――――――― 2 2 ―――――

= 3 = 2 3

―――――― ) ) x x ( ( 1

Elaborate The divisor is a factor is 0. of the dividend when the remainder The The divisor must be a linear binomial with a leading coefficient of 1. The dividend must be The of 1. a linear binomial with a leading coefficient divisor must be The missing terms. any with 0 representing form in standard written If one linear factor find the division can be used to of the polynomial is known, synthetic be easily factorable. product of the other factors, which may The numbers generated by synthetic division are equal to the coefficients of the terms of of the terms the coefficients equal to division are synthetic by numbers generated The the same process. essentially are They including the remainder. the polynomial quotient, p Essential QuestionEssential Check-In

What conditions must be met in order to use synthetic division? use synthetic to be in order met must conditions What How does knowing one linear factor of a polynomial help find the other factors? find the other help a polynomial of factor linear one does knowing How

Compare long division and synthetic division of polynomials. of division synthetic and division long Compare p

-4

Your Turn Your

the remaining factors of p of factors the remaining 9. Module 6 13. 12. 11. 10. Determine whether the given binomial is a factor of the polynomial of a factor whether p is Determine the binomial given 8.

A2_MNLESE385894_U3M06L5.indd 328

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CorrectionKey=NL-D;CA-D DO NOT EDIT--Changes must be made through “File info” “File through be made must EDIT--Changes DO NOT CorrectionKey=NL-D;CA-D DO NOT EDIT--Changes mustbemadethrough “File info” esn 6 .5 Lesson 329 1. leadingthe coefficient divide dividend the and divisor by aconstant to make It method. best the may, however, possible be to binomial with 1,long leading coefficient division is Point out that for adivisor other than alinear synthetic division and to long when use division. Students may about confused be to when use for synthetic substitution and synthetic division. division. Remind that them terms are always added values are rather added than subtracted as inlong synthetic division and synthetic substitution because Students might make errors insigns doing when ASSIGNMENT GUIDE AVOID COMMON ERRORS AVOID COMMON ERRORS EVALUATE Factor Theorem Using theRemainder Theorem and Example 3 Synthetic Division Dividing p Example 2 Division Dividing Polynomials Using Long Example 1 Using Synthetic Substitution Evaluating aPolynomial Function Explore Concepts andSkills ( x ) by x-aUsing by Practice Exercises 12–15 Exercises 9–11 Exercises 5–8 Exercises 1–4 CorrectionKey=NL-A;CA-A DO NOT EDIT--Changes mustbemadethrough "File info" A2_MNLESE385894_U3M06L5.indd 329

Exercise © Houghton Mifflin Harcourt Publishing Company 16–19 1–15 23 22 21 20 3. remainder. Write result the form inthe dividend long use to division quotient the andfind dividend, divisor Given and apolynomial 1. Given oue 6 Module 6. 5. You out acheck. may to wish carry 5 -3 2 -3 87 2 4 p p 2

p -3 73x

p ( x 18x ( x

-17x Evaluate: Homework andPractice 6 18 ( ( ( ( 2

2 p x x x

-3 -3 4 - ―――――――――――― - 8 + ) ) x 4 (

x 3 + = 2 = 8 ( ――――――――― - ) - 3 3 3 3 2 2 1 Depth ofKnowledge (D.O.K.) ) ) 6 , find p ,find

⟌ ⟌ x Strategic Thinking Strategic Thinking Strategic Thinking Skills/Concepts Skills/Concepts x = 0 = ( ––––––––––––––––––– –––––––––––––––––––––––––

―――――――― 4 51 -24 ―――――――― -6 3 0

8 8 x x 3 Recall ofInformation Recall 18 18 ―――――――― - 6 4

6 x x

3 3

- 9x -155 x + 0

2 ( + 5 + 7 x x 4

――――――――― -

2 -3 +

7 53 -17

――――――――― - 3

3 + + -1 0 0 - 3

( - 3 + 0 ( x x ( x x x x + 13 -3 x x x

3 x -48 -48 2 2

-1

2 2

+ 48 - 3x + 2x - 48 + 0

- 48

+ 0x + 73x x x + 0 ) )

2 2 by usingsubstitution. synthetic 0 -3 )

x x ÷ + - 72x ÷ x 2

x x + 4

2 x + 0x - 17x x (

2 - 13 + 12 ( + 8x

x ) - 9x x

- 1 1 - -159 -155

- 9x 2 2 - 4 + 8 )

- 384 ) + 397 )

+ 13

) 3 13 +

)

4. 2. p -3 16725 5 -3 1 0 p

p p ( ( ( ( = = 6 = 6 ( Check. = 18 = 18 ( Check. x x -3 329 -3 x x ) ) 2

(

2 = =

――――――― -3 divisor

MP.2 MP.6 MP.2 MP.2 MP.4 MP.2 + 8 x x 3 1 - 4 ) ) COMMON

4 4

- x x x = =

―――――――――――― CORE

+ + 1 8 -24 64 -1 3

3 3

x )

) + 6 -19 - 3 - 72x -147 ( 4

Reasoning Precision Reasoning Reasoning Modeling Reasoning ( x x

6 + 5 18x )

3 3 x

( x - 9x - 48 6 -9 x -19 -2 2

quotient 2

Mathematical Practices

2 x + + 7x

- 3 3

+ 3 - 3

- 8x x x x + 13 - 48

2 )

- 1 -24 72 x - 25

+ 48 + 73x

2 ) + 45

+ 1273x +

)

- 17x x remainder

2 - 13

8 45 -8 + 8x + 397 - 384 -192 - 13 . -147 • Extra Practice Extra • Help and Hints • Homework Online •

17x + 397 esn 5 Lesson 3/19/14 11:57AM 330 Dividing Polynomials AVOID COMMON ERRORS COMMON AVOID A common error when doing synthetic division is to to is division synthetic when doing error A common Show it. adding than rather row the second subtract its alongside problem division a long students that emphasize to division synthetic using solution, if they this error. make will be different the results DO NOT EDIT--Changes must be made through "File info" info" "File through made be must EDIT--Changes NOT DO CorrectionKey=NL-B;CA-B 6/27/14 7:01 PM

x - 3600 Lesson 5 + 3600 - 2x + 0.5 - 9 - 1 - 6.75 - 7x + 9.5

+ 18x 2

) x - 9x - 9918x

- 18x + 25.5x - 128 + 28 )

- 2 2

) x - 28 - 1

) + 3.5x x

- 9 + 400 + 16x

+ 8.5 3

- 28x - 6.75 x + 10, 000x - 10

x + 10 2

2x 2

( x + 28 x

- 0.5 )

) - 3x 3

+ 3

2 + 3.5 x 2

- 2

- 9x

- 416x

x 2

x + 9

3 - 100 2

- 5.5x + 50 + 100x ( 2

+ 3.5x - 9

x x 2

) 2 2

+ 16 x 2 - 2 - 3600 8

x x x x 3

(

and obtain obtain a and x ) + 8.5

+ 25x 2.5 + 2.5 + 6 3

- 2.5 x + 4.25 - 18 + 52

- + 0.5 (

2 3 3

x 2

- 400x ) - 28.5 + 2 + 52x

x x + 250 + 200 x x 2

1 4 4

2 3 3

1 2 x _ x x x - 3 - 9918x x x

( + 0.25 + + 6x

8 - 0.5 + 9.5 7 + 3x 4 4

( x x 2

x - 1.5x x Check. = = ) - 56 - 4 ( ( = 2.5 =

= 2.5 = 8

3 3

( = = Check. x x

- 8 )

330 x

( = 7 = 7 ) 10

9.5

-0.5 ) x - 3600 + 3600

- 2.5 + 0 ) - 2.5 7

+ 4.25 ) - 6.75

-9

-2

)

) ) + 0 + 9 + 6x - 1.5 )

28 9 2

+ 6x ) 128

-1 x x 2

- 8 + 1 + 7.5x -9918x + 0.25 -18x -100

-10 . You may wish to carry wish to may a check. out . You + 100x x x x

2 ) x 0.5x

( 2

+ 18x

( + 25x x

(

+ 25.5x a + 82x - 9

+ 10, 000x

x 2

+ 0.5

2 -28 (

2 + + 0 ÷ x ÷

- 0.5

÷ p x

+ 8.5 x x x 3

-28.5

16 8.5 3

) ) 1 2 3

-9 _

) 416 x x

( x + -3.5 -5.5

8.5 + 3x -400 -3

) , use synthetic divide division to x by + 400 ( 200 200

2 + 250 + 50 ÷ (

)

( - 10 -3 + 3 x + 0 - 3x

3 ) + 10

- ――――――――― x

0 - 100

( - ―――――――――

4 2

2x x x ( - ―――――――――――

––––––––––––––––––––– ( x x x -2 0.5 -2 52 ⟌

–––––––––––––––––––––––– ÷ -4 (

⟌ - ――――――――――――――― 3.5 ) - 9x - ――――――――――― -2.5

- 5.5x

2 quotient 8 + 100x

2 x ( 56 ――――――――― ―――――――――――――

- 400x

) x

x - 2.5 + 0.5 a

―――――――――

7 8 + 6

3 + 25x + 9 - 28.5 - 4 x

x + 250 + 6x

2 2.5

4 3

( + 3x

8 7

3 4 x x

x 2

7 x x 2.5 8 = 1 2

-1 2.5 6

-0.25

( ( ( ( (

) x

(

10. 9. Given a polynomialGiven p 7. Module 6 the quotient and the (nonzero) remainder. Write the result in the form in the form the result Write remainder. the (nonzero) and the quotient p 8. 11.

A2_MNLESE385894_U3M06L5 330

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© Houghton Mifflin Harcourt Publishing Company Module 6 Module 15. 14. Determine whether the given the binomialDetermine is p whether afactor of polynomial the 16. If so, remaining the find factors of p 12. x 81

So, So, 3 This gives theexpression 1 This gives theexpression x 3 p p coefficients is modeled by is modeled coefficients V function the The volume of arectangular prism dimensions whose are binomials with integer of prism. the Given that x that Given x x So, So, -2 1 p

( (

x x ( + 2isafactor. 4

- 14x

1 1-40 x

) ) - 1= p 1

) p = 4 = 1

( = ( x x ――――――――――

x ―――――

――――――――

―――――――― ) 1 1 4 x ) x ―――――― 3

- 22 + 45= = =

( + 2 -2 - 12 -7 -8 x -12 -7 x - 1and x

x ( 0 2

3 + 1 x 3 12 1 -22 -

x - 22 x

0 + 2 + 1 2

x 8 14 2

-1 0 -12 -1 - 2

+ 157x )

-7 + 2x

( ( 0 12 19 12 x ) x x x x 2 0 2 (

2 360 -112 - 5

2 - 1

x -

157 + 157x - 3are two of dimensions, the missing the find dimension -2 - 1 - 5; 45 2; x - 360; 2 -12 ) -5 )

( - 2

(

) 12 6 1 x x ( x ( 0

-360 + 2 - 9 x x

- 4,whichisthemissingdimension.

- 3 -7x + 2 - 360= x 0 ( ) - 3isnotafactor. ( ) x

) - 8 )

)

. + 12. . x - 8isafactor. ) (

( ) x

= - 5 x 3

331

- 8 ) ( 13. x x - 9 2

+ 19x x -2 p ( ) + 2isnotafactor. x (

) x

= 2 2 - 8 - 12. ―――――――――― -4 -4 x ) 4

. + 6 ( x x 6 2 ) 3

. - 5x -4

0 - 10; -5 3 8 ( x + 2 -16 -10 -6 )

Lesson 5 Lesson 3/11/16 2:45AM 332 Dividing Polynomials Emphasize the conditions that must be met must that the conditions Emphasize Point out that students should review how to to review how should students that out Point INTEGRATE MATHEMATICAL MATHEMATICAL INTEGRATE PRACTICES MATHEMATICAL INTEGRATE PRACTICES MP.2 MP.1 MP.1 Focus on Math Connections on Math Focus on Reasoning Focus do synthetic division and long division when the division long and division synthetic do the of powers some for terms missing is dividend the include they must that Emphasize variable. the of the times power zero as written terms missing the division. complete to variable to use synthetic division: The divisor must be a linear must divisor The division: use synthetic to dividend 1. The coefficient a leading of with binomial 0 representing with form in standard be written must terms. missing any that division synthetic of Discuss the characteristics only variables, no are there synthetic: it make of used is instead addition coefficients; and subtraction. info" "File through made be must EDIT--Changes NOT DO CorrectionKey=NL-C;CA-C 6/9/15 12:07 AM

2 - 8 x -

3 - 2x x

3 x - 1 .

-8 B. + 3x

2 x -2 12 -20 0 332 1, use polynomial division to find to division + 1, use polynomial 2 and the volume is is the volume + 2 and -6 + 5t.

> 0 2

t ――――――― - 3x. 3 -6 10 -28

x -2 3 . Write an an + 10t. Write

2 t

. + 6 2 , where t + 2 , where .) )

3 t

. Given the height x the height . Given Bh

0 0 0

0 0 0 1 3 =

1 3

- 1 __

) 0 t

- ( = = 0.5 -1

) 20 10 t 2 3

-8

__ 6 0 + 2x ( Two students used synthetic division to divide 3 divide to used division synthetic students Two

2 2 -10

+ x

. -1

2 -3 1

l 5 0 6 P

__ x -2

4 3

__ 3 4 + 4 =

A. The volume of a hexagonal pyramid is modeled by the function modeled is by pyramid a hexagonal of volume The

3 6 12 0 + -1

x 3

-2 -1 -6 -3 0.5 1

(

A Van de Graaff generator is a machine a machine is generator de Graaff A Van x

―――――― -1 ――――――― 1 3 _

1 3

__ 1 1 =

―――――――― =

1 0.5 2. Determine which solution is correct. Find the error in the other solution. in the other the error Find correct. is - 2. Determine which solution

) ) x

x 2 3

x ( (

3 6 10 12 So, the area of the base can be represented by by of the base can be represented the area So, by can be represented the area So, Student A is correct. Student B used the incorrect sign of a. incorrect B used the Student A is correct. Student V -2 -2

The voltage can be represented by 0.5 by can be represented voltage The -1 represents time in seconds. The power of the system the system of power The time in seconds. represents P be can modeled by Explain the Error Geometry Physics Given that the height of a rectangular prism is x is prism a rectangular of theheight that Given by expression that represents the voltage of the system. the system. of the voltage represents that expression V that Recall write an expression that represents the area of the the base prism. of of the area represents that expression an write an expression for the area of the base. of the area for expression an V a pyramid, For (Hint: V that produces very high voltages by using small, safe small, safe using by very high voltages produces that a current has machine electric One of levels current. l be can modeled by that

19. 20. Module 6 Module 17. 18.

A2_MNLESE385894_U3M06L5 332

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info" "File through made be must EDIT--Changes NOT DO

CorrectionKey=NL-D;CA-D DO NOT EDIT--Changes must be made through “File info” “File through be made must EDIT--Changes DO NOT CorrectionKey=NL-D;CA-D DO NOT EDIT--Changes mustbemadethrough “File info” esn 6 .5 Lesson 333 help remember them steps the insynthetic division. Have students amnemonic describe that device can havethey for one or method other. the have results their discuss them and any preferences polynomial that not does have alinear factor. Then exercisethe for division anew problem with a division. Then have students switch roles and repeat other the studentwhile solves it by using synthetic polynomial division problem using long division, Instruct one student ineach pair to solve a JOURNAL PEER TO PEER DISCUSSION CorrectionKey=NL-D;CA-D DO NOT EDIT--Changes mustbemadethrough "File info" A2_MNLESE385894_U3M06L5.indd 333

© Houghton Mifflin Harcourt Publishing Company 21. Module 6 Module 22. 23. H.O.T. Check. = 3 = = 3 The quotient needsto bedividedby 3. ( Rewrite 3x - organize. numbers, integers, or rational numbers with to respect closure. Use table the to under four the basic operations, and determine polynomials whether are like whole operations. Then consider of set the polynomials whether inone variable is closed of integers, and of set the rational numbers are under closed each of four the basic by by Critical Thinking Multi-Step Analyze Relationships the constant terms ofeachfactor. of thex-terms ofeachfactor. The oftheGCFand value ofdistheproduct a 3 They are closedundereachoperation except division. Polynomials are similarto integers to closure. withrespect __ 3 3x ( nonzero) Division (by Multiplication Subtraction Addition x = 3;d _ x

3 4

( (

2 Focus onHigherOrder Thinking 3x - 2

x x

3 + 4 - 15x

3 3

+ 4 - 11 + 4 + 4 ) 3 ) ( = 72; The oftheGCFandcoefficients value ofaistheproduct 3 -11 x ( ―――――――――

) __ x 3 + 3 ) .Then check solution. the - 36

2 x ( x + 4as3

Use synthetic division to divide p x

- 15

2 2

) - 56x - 15x

-15 ( - 5x x -4

= - 4 3 x The polynomial a polynomial The

Numbers

2 x

( Whole - 20x - 12 ) - 50 - 36 x 2

.What are values the d?Explain. of aand

Yes Yes No No - 5x + -56 -36 Investigate of set the whole whether numbers, set the 20

_ 4 3 )

)

- - 36x )

- - 12 . -50 2 2 -2 48 - 482

x Integers

+ Yes Yes Yes No b x 2

( + x ) 333

cx = 3 + x d is factored as Numbers 3

Rational Rational

- 11 Yes Yes Yes Yes x 2

- 56x Polynomials - 50 No Yes Yes Yes Lesson 5 Lesson 3/11/16 2:45AM

334 in the Dividing Polynomials . Ask them if the remainder is is them if the remainder (t). Ask 3 such groups, or divisions, for for divisions, or groups, 3 such avg there are Have students highlight the highlight remainder students Have AVOID COMMON ERRORS COMMON AVOID MATHEMATICAL INTEGRATE PRACTICES zero or nonzero and what a nonzero remainder remainder a nonzero what and nonzero or zero a discuss the significance of students Have means. describes attendance a function that for remainder ask per team, 315.8 fans is the attendance If per team. if it actual or person an if the 0.8 represents students the model. of a limitation from results Students may try to include the “3 divisions” in their try the“3 divisions” may include to Students teams of the number dividing by perhaps calculation, a division that students to 3. Explain by attendance or other each against compete that schools of a group is and that is this number that Explain basketball. college is important is What this calculation. for irrelevant the total represent in the table the numbers that all of for attendance the total and teams of number basketball. collegiate women’s MP.2 Focus on Reasoning Focus final function A Scoring Rubric his/her reasoning. and explains the problem solves correctly Student 2 points: but does not fully good understanding of the problem shows Student 1 point: his/her reasoning. or explain solve understanding of the problem. does not demonstrate Student 0 points: 3/1/16 1:21 AM

) where t ) where

(

10,878.3 11,120.8 11,160.3 11,134.7 11,160.0 11,201.8 Attendance Attendance all 3 divisions by T . Carry out the division to to . Carry the division out

) ) ) t t t (in thousands) for for (in thousands) ( ( ( + 1005 T A

___ =

) t ( 12,981,600 avg

10.57t __

attendance (in thousands). = attendance A 334 . 1003 1013 1032 1037 1048 1055

) t ( T in all 3 divisions + 10,880 Number of teams Number of teams remainder ______

+ 12,927.9 - + 329.8t

t 0 1 2 3 4 5 ) and cubic regression on the data pairs (t, A pairs the data on regression cubic ) and number of teams, and A and teams, of = number for the same year. Ask the students how accurate the model accurate how the students Ask year. the same (t) for - 135.64t

avg Years since since Years 2006–2007 t - 121.6

+ 1005 3

NCAA Women’s Basketball Attendance Basketball Women’s NCAA t = 1.30558

) quadratic quotient quadratic in the form = 13.80 t = 10.57t )

( t ) ) ( t t avg

(

avg ( Season T A A A 2006–2007 2007–2008 2008–2009 2009–2010 2010–2011 2011–2012

EXTENSION ACTIVITY EXTENSION is in predicting the attendance for that team and what some of the sources of error error of the sources of some what and team that for the attendance in predicting is be might Have students research the attendance for a specific NCAA women’s basketball a specific women’s NCAA for the attendance research students Have calculated the value to er numb that them compare season. a single Have for team from the model A

write Use an online computer algebra system to carry out the division of A of carry to the division system out algebra computer online an Use Module 6 Models: The table gives the attendance data for all divisions of NCAA Women’s Basketball. Women’s NCAA of all divisions for data the attendance gives table The Lesson Lesson Task Performance

Online computer algebra system result: system algebra Online computer Enter the data from the second, third, and fourth columns of the table and perform linear linear perform and the table of columns fourth and third, the second, from the data Enter (t, T pairs the data on regression since the 2006–2007 season,since T per team: attendance the average a model for create Then

A2_MNLESE385894_U3M06L5.indd 334

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