The Remainder Theorem of Polynomial Functions

Let's review from Algebra II how to synthetically divide a polynomial by a linear divisor.

1. (-2x2 + 2x -3x3 + 1 )+ (f-1)

3 t>. ·-3). -):x 1" ':b. -t \

Perfonn synthetic division

2. 3. (3x4 +x2 -3x+"3)+(x+ 2) (2x3 + x2 -l3x+ 6)+(2x-l)

Perfonn synthetic division Perfonn synthetic division ')(.-, =o � ')(.-: \ ... �--t'4� 0 -'7 X-==--:l :2..�-\ /0 ...i., 2.x.�, � X:: L

ffi -?, -.l 'l.. \ �3 0 \ -3 3 � !l \ -\!) Ca

o· -3 -� -� -.2\o ., -Ip\:). 0

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-3�').

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.

The following activity is designed to show you what the � of Algebra states.

For each of the functions in the table below, use the graphing calculator to determine the given function value. Then, use synthetic division to determine the remainder when the function, P(x), is divided by the given divisor.

.

Function Divisor Remainder when P(x) is divided by the

iven divisor.

If P(x) = x3 + 2x2 -x-3, find the value of [D \ 2 -\ -3P( l). 0 ,, ., 3 (x-1)

,-, �-\�o \ 3 1 X:=-1

��� -\

If P(x) = 2x3 -x -3, find the value of P( 2). � � 0 -, -3

?0-)� \\(x-2) D :\ ''3 \�

x-� -::.o � '1 ���

::. \ I

If P(x) = -2x4 + x 2 -x -:1; find the value of D \ -l ·-l

P(-1) (x + 1) .). -� \ 0

,X-t\� :;;). -\ 0 [ j x.-=--1

Function

If P(x) =2x4 +7x3 +4x2 -7x-6, find the value of P(-f ).

�C-f)� o

If P(x) =-3x2 +x-1, find the value of P(2).

Divisor

(2x + 3)

�.xt3 ::;0 l)..)(= -3 x-=-�

(x- 2) x-;l:::-0

)(..-::.;}.

Remainder when P(x) is divided by the

.

iven divisor.

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�=O

� -3

-�

\ -\' -� -lO,-,_ i::_. \I I•• --� c.!l

\<�::: -\\

-tolo '

� �� � Q_� � =T\C:.... Cf\.kv�t.»\<i'f'.. ,�,� What do you notice about the five previous examples?

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� o..c; � j-v� i �()'r-. M � +t, � ?-.--v'� �� � *� J.lvts� ��'

1. For what value of k will the function P(x) = -2.x3 - 2.x2 + kx- 2 have a remainder of 8 when divided bythe factor (x + 2)?X-r.?.:=-D [ -,.fil - � -� K.. -�

X-:::: -:J- e) � - L\ -��+g� .. ���.:-:.����--��-.--F-��-·+�-- -��-::::2,

LK=-,-l

2. For what value of k will the function P(x) = 3.x3 + kx2 - 5 have a remainder of 4 when divided bythe factor (x - 3 )? ��-;�o l]J 5

x�3

q \{ + rtb =--1 q lL:=--ttlJ

}1<-=: -a)

3. For what va1ue of le will the function P(x) = -x4 - 2x2 + kx- 6 have a remainder of O when divided by

the factor (x + 1 )?(X4U::. 0

X:: -\ , -\ 0 -\

0 t

l

-�-,

-:,

-b

3 -K-3(Kt3) \-K-'l

-"'-C\ -=-0

-�='((\L=-'f)

4. Suppose the function g(x) = 3x2 - 2x� + 3x-2 is divided by the factor (x - 2). Which is greater: thevalue of g(-2) or the remainder when g(x) is divided by (x- 2)? Show your work.

ffi ,.!:)_ 3 3 0 -i -�

Th.� equations of the functi6ns below are given in both factored fonn and expanded fonn, and a graph of the function is given, as well. From the graph, identify the zeros of the function. Then, synthetically divide the expanded fonn of the function by each factor.

1. j(x) = (x + 2)(x - 1 )(x + 1) f(x) = + 2x -x-2

m \ ;).. -\ ·o \ -�� \ � -\

0 -), 0 \ 0 -\ \ 3 :)..

B \ � -\ -). 0 _, --, �

-\ \ -� ).el_

2. p(x) = 0.2(2x + 5)(x - 1 )3 p(x) = 0.4x4 - 0.2.x3 - 1.8x2 + 2.6x - 1

O.l\ -0.� -\/! .2.Jo -\ 0 -\ � -3

� o. L\ -\. d, \.).. -0, "'\

ffi o.L\ -0 .. 2. -\. '& �.to -\

0 0.""' 0 '')... '-\ .. \.:, \ e.L\ ()/}... -\.� \I� \Q_

!········?········1········?''''''''!''''''''+· .. 5· : : . . : . i ........ i ........ i ........ L ..... i ........ i .. : .: : : : : : i ...... l ........ i ....... l ....... .i ........ i ...: : . : : . .

,.,,,,<,0••-•••I •••••O••••••••I••••••••+•••

..... .v\-1- , , , , . . l 1 \ 1

...... +·· ·····!········+········!········t········ii \ I I t . . ...... : ·······j········t········;········:········:

.... A ....... � ....... t ...... 1 ....... 4 ....... 1J.1... ..... 1 ..... ) ....• -1--..... i ... .1· · · i 1 .! l 1 1\·······+·······\·······+· .... ; ........ f ... 3· + i + · ,o, =

i ....... i ...... .L.. ..... } .. L ..... i�. :::::r::::L:::c:1

·1::::::i::::::::;

\(-.::.- ,-,.�

: : l : (·····t······:········t ··f···· .. t· .. 1·i·· ··(······i······(··· ·}·(·+ : l : : ; : � 5 -4 --S i h ······ ,,, .

... .:i',,.· .... J',,.

' ....... 4',,.' ... )',,.

·······�,,:'· : ···i .... 1 .... i .. ·1 ····: i·

!········t········i, ········t···. .. !········+ ... · ······!········!····· ···t········l··· .. ···+········l ...... !,, ..... ),, ........ l ...... J ....... , . ...... ..l i j j j i i . . ' . . ' (""'+ .. ······\""·"·(···· ·�··· .. ···+ ······+········i···· .. ·+······+······t·······\

11 l F \.�; ll r l r I lC-=- -�.1, � i.

Answer the following questions about factors and synthetic division.·

1. Is (2x + 1) a factor of g( x) = 2x3 + 3x2 -1 lx - 6 ? Show and explain your work. What is the

graphical interpretation of the result of this work? �)("t\ =-0 \:._g � 3 -\ l -b .,

/J.1'.-:..-\ .> 0 -\ ..:, (o �-:. -v-,.. f>- . ;,.. _, d- l1i..

c_.,__,..-t \) \ � :· CL� 1) 5<x) t>\c.. � �� I� (J'S, lr\u.. �� -::: o , �J(-i) --::.o so� 8�4 gU') , <; °" *.Q_ �-�s. �-=- -k, 2. Is (x + 2) a Yactor off (x) = 2x

3 -x

2 + 3x-8? Show and explain your work. What is thegraphical interpre

,tio

uf the result of this work?

&-t-.).)�6 --:). " -\ "; - �)(-::: -:)... 0 -� h) -.i:.'li

� �5 \3 l3i. ()<. ;-2) \'> V\.Pt 0.. � 1 f!.(,c) lo \c -tl.. � .:t O.

�nc..e- � � :=-?a'-C J -tt,� «_-2.)-=.-3'-\ L.O"?D .4k.- 'j� � -f-c ... ) ls -..� -t\.u'. l)(i -�s .J. ��-

3. Consider the functionj{x) =? - 13x + 12.

A. Divide j(x) by (x - 3) and B. Divide fix) by (x - 1) andthen factor the quotient. then factor the quotient.

)'.-1:=-o )t:. .s >t.-\ �o )(. .. \ , o -,i ,i CD \ o -\'I

a -,2. o , , �i:---------"'!-..... � , a , ' -\'2..

�r.+ax - '-\ x�+�-,i.

lX-\)(X�'i) �-+'1)(��3) '!

C. Divide fix) by (x + 4) andthen factor the quotient.X �4 :. O ><. ::"'-,t

0 \ -'1 3

�� t-0 \ e _,, � 0 -� -t<-

��'tx�3 c..�-3)�-\)

t�: l')(-i)�-\�� .. � �()t.)':. (�-oc�-t"t»-3 �(>c.):: (�-t\.QQ.\-tlx-t

4. Based on your work in exercise 3, w�at are th�zeros of.f{x)?�()..) � c-x��(x-,J�- i.;

>t-:-� ) ,,�! 5. If (x - 1) is a factor of g(x) = 2x3

- 3x2 - 2x + 3, what are the other two factors of g(x) and rew1ite

g(x) in completely factored form.

>c.-,�o ffi � - 3 -1 3 -3 ��, � � -- ��--.;...,---3-.,.[Q.-:o

�'t>'l. - X. - 3 � �<1-):: (?-,c.-;) (x + I )(x.-1�

6. If (x + 2) is a factor of h(x) = 2x3 + 3x2-8x-12, what are all of the zeros of h(x)?

a. _, -Co�-�-1.c

h�)= (.�-\'3 )(..,c..-�(x+i) 7. For what value of k is the factor (x-2) a factor of the functionp(x) = 6x3

- 19x2 + kx + 6?

Cf) b -,9 � -':, ��-1'2..:: o 0 \ 2.. - \ '{ �"'-� :l\L-:. 2.2... b - '1 �-l'i} l8'11..-:t>- ( �-:. \I ]

8. For what value of k is the factor (x + 3) a factor of the function q(x) = -3x3 + kx2 -23x- 6

c�} -3 K -�� -Co q_�+\�'-l-=-0 0 G\ -3�-l'\ Q.k-+\Sb C\,\l-- --\� -� � c_3�-6t)) ��H-.1..\

\�; _ \t.j

9. Is (x + 1) is a factor off (x) = -3x4 - 7x3 -3x2 + 3x + 2? If so , how many times is it a factor?

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-'1 3

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( ! x-:2.)

Name_....:...�-=-=-����-�,;___;_-=-t-________ Date _________ Period \ �-

Day #16 Homework

I. Suppose the function f (x) = -2x3 + 2x-:- l is divideq by the factor (x + 2). \Vhich is greater: thevalue offi:-2) o.r the remainder whenfi:x) is divided by (x + 2)? Show your work.

w� ?t-t,)..�o J � ?(:-l. �CC..OrtL.� '1t) �·. �kt­�� f(--i) = � � �-P-uc)�(x.+�.

l• Suppose the function g(x) = x2 -x3 + 2x-l is divided by the factor (x- 1). Which is greater: thevalue of g(-1) or the remainder wheng(x) is divided by (x-1)? Show your work. rn-\ \ .2 -\ .3c-u � (-i)�(-\)3��-,)-\

-, 0 2

- l b 2. ll jl-\)-:. \ -t \ -1-\

'K��::. \ �t-\J -:. - \ �, For what value of k will the function h(x) = -x3 - 2x2 + kx-4 have a remainder of-2 when divided

by the factor (x - 2)?

@ -\ .-�. -�

-\ -4

!l..\:'--10 : -;i

;l�:: \�

l�=:9J

J.f- For what value of m will the function g(x) = mx2 -3x + x3 + 2 have a remainder of 2 when dividedby the factor (x + 1 )?

BJ \ � -\

\

-3

-Mt\

VV'I +-L\ -:: �

,�=-;lj

"5'. For what value of m will the function p(x) = 2x4 + mx -4 have a remainder of O when divided by the factor (x + 2)?

l:� A 0

-�

0

�

M-\G -d,-M=-�S

�M� \1]

• I \ /

- -For problems b - '\ , use synthetic division to determine if the given factor is a factor of the function

J(x) = x3 + 3x 2 -1 Ox -24. Show your work and write yes or no.

4. (x+4) 'J, (x-3) g�., (x + 6) 9 .. (x + 2) � I '?> -\t)

�

ffi\ ; -\0 -i� 61 \ 3 -,o �� rn \ 3 -� '1 ; l& � -b \& -�a. -;_ \ -, -(, b & \ -3 8 tn \ \

je.S> �e& NO �

-IO-,l

-,i

fO .. What reasoning did you use in exercises 9 - 12 to detennine if the factor what a factor of.f(x) or not?

""If � �� -:::oJ

� � c:U.l,sor ,c; o.....

+c...e,� � -t'6'-).\I, (x + 2) and (x -3) are the only factors of the function g(x) = x3 + x2 -8x -12. How many times is

each a factor of g(x)? Show your work.

t-:>.] \ l -s - '2-.l. � l*

\ -\ x'). -x - b

(_X - 3 )� + ;>.) l.l. If x = 3 is a zero of the function h(x) = -2x4 -63x2 + 19x 3 + 81x -27, what factor is a factor of h(x)

and how many times is it a factor? Show your work.

-�� �

:r.f "� 3-i� � � J ro-1. l'\ -L,3 8, -::>, �"(( � ,"\ 1p l9 -1:,.. ;;l1 �� w� 0 � tx.-�� ,� � -;>,.. -,3 --1'-t � lsi. '3 ti�.,� --,r-...,\e.- 1> \·b.J . rn -)_ -l, :3 Tu c.-x - �) t., ti...

� -i [fl � ,J h(.>.)CID -). -L 5. -bi��

·" -.)_ [:S /3. f (x- 2) is a factor of J(x) = 2x-' -3x 2 -3x + 2, what are the other two factors? Show your workusing synthetic division.

ro� -3 -34 �

9- )(.ik \ "?<.. -\

(2.-,.. - \ ;("X + \)

[Q_

�,c.-0 � (�,;-i) ��

0� � ti"-'�i �-

1&;\ Using a graphing calculator, draw the graph off{x) on the axis provided below. Then, explain why your work in exercise 11 is validated by this graph.

If� 1)� C-x-�, (4-,),�(_ "t. --t \J c,.A-(2_ � = o � � <5"\vecl

)

� �::-\, �, � .i �� J\.t. � u \. +; V) � � 0-A-t. �

�s � � 5v+ �-;k_�c. -\-iO'r\ .

-1

-l

-1

-, 2 15', For what value of k would the factor (x + 3) be a factor of f(x) = -3x" + kx + 20x -12? Show your

� �-3

-3

-rl...9K tl.)

Cf

""' +'\ = 09\l-��

\ �=-lj 4 " 2 j(.. For ·what value of k would the factor (x - 1) be a factor of g(x) = 3x + 2x" + kx. -2x + 1? Shov-,,

your work.

\D 3

s

K s

-�

\l\-S

\L '\-'t "=- 0

, [L�-tJ