Zeros of Polynomial Functions

Analytical and Graphical Approaches

For each of the functions graphed below, state the end behavior and the zeros of the functions. i········r········i········r········:········t···S· ······0···-··+·······-0-········l········,O.········· . . . . . .

. . l ........ 1 ....... i. ....... 1 ........ 1 ........ l.: : : ; ; : ;

: : : : : : l? j ........ t .. ······!········t········l ........ t ... i,: l : : : :

J;; � � -B 1••••••••0••••••••1••••••••,0.••••• •l••••••••O .

, ... L . ... L ... \ ...... .t3. . . . . . . . . .

.... r .... l .... J ...... t ... A .....i i ·l i ...... +········l········+········l········+········l !········+········!········+ ····!········t-3·

.... , ...... . ,-4 . ...... ! ....... .!... ..... ! ...... ..!.. ...... : .... ....! Graph offlx)

f{x) is a cubic function.

Left End Behavior

k:, � .... -ce> �l)j ... -co.

Right End Behavior

At:. � .a,, 00 )

-+(�) � co. Zeros

···r·······:·······"1··'4· : l ! : ' ...... ( ····'··· ····t--·3

:··· ··}····--;--·····;· ·· ··,.'········:.

' ·· J·+·····i·······i·········;·I····'········ '········' . . : ! : !········'!'········!········+·· ·····!········'!'···l·-'-······•········•··············t·······t········i I l : l i I ! 1 �--t4--i�� r·······r·······T·······r .. ·· ···1····· ... ; ... 1·

. .

·V. ;J ······+········!········t···"·"l··· .. ···+ .. ······1..... ;, ........ :.,\ .. :.,-'· . . . .... L ...... L ...... .L ...... .J.--<i . ............... '. ........ '. ....... .i. .... ;

Graph of g(x)

g(x) is a quartic function.

Left End Behavior

Right End Behavior

� �-.oe)

at�)-. oo. Zeros

... : ........ : ........ :. 5·-····················································' Graph of h(x)

h(x) is a quintic function.

Left End Behavior

� 'X. .. - QCa.)

V\ u-.) -", -�. Right End Behavior

k �_., CDJ

\'\{)(.) ... c.Q. Zeros

')(::. "-- 0.-0 � � "<

(�-,a., \� �-kc..� ·an.Read the following information about the multiplicities of the zeros of fix), g(x), and h.(x) while studying the graphs above. Then, answer the questions on the next page.

In the graph of fix), all of the zeros have a multiplicity of 1.

In the graph of g(x), the zero of x = -2.5 has a multiplicity of 1 and x = 2 has a multiplicity of 3.

In the graph of h(x), the zeros x = -4, x = -2, and x = 5 have a multiplicity of 1 and x = 2 has a multiplicity of 2.

1. What do you notice about the sum of the� � :. � � u 'IV\.

multiplicities of the zeros and the degree of the� ..... 0. .. �� � e() ,.,,,d_ � function? , • ._ -o

2. Describe the behavior of the graph as itapproaches a zero whose multiplicity is 1.

3. Desc1ibe the behavior of the graph as itapproaches a zero whose multiplicity is 2.

4. Describe the behavior of the graph as itapproaches a zero whose multiplicity is 3.

.

�\

For each of the following functions, you are given a factor of the function. Identify the zero associated with the given factor. Then, use synthetic division to detem1ine how many times the given factor is a factor of the function. Then, identify the multiplicity of the root associated with that factor.

fix)= .x3 + 2x2-x-2

.. J .... d ...... ; ....... f

h(x) = 2x3 -x2-4x + 3

Factor: (x + 1)

@ \ !l. 0 -,

\ \

Root: )(:. - l-\ -i 8) \ \ -�-\ � 0 -\ V -� � �,--o��\..:?:�-�

Multiplicity; :l Describe the behavior of the graph at this root.

'L ��

w,�

Factor:

1 A ¢

(x- 1)

-,

,.

Multiplicity: �

Root: '1'. : i

-4', \ -i

3 a.

\

!2. 3

Describe the behavior of the graph at this

p(x) = -x4 + .x3 + 3.x2 - 5x + 2

:···········:···········:···········:···········:·····9-

1···········1·········+·······+··· ···· '·····S · ........... · ........... , ........... : ... ...... l . .... -1 ........... !···········

1 ........... 1 .. ....... ! ... .. �

i : i= :L��i :=j·· i···········i···········i···········

1 .......... j ..... :

i···········i···········i··········· ···········i······l·

. . . '···········'···········'········· ············'··· 3·

! ! ! ! ! ••..••••• j •••.•.•.••• j ••••••••••• j ••••••••••• j ••••••••••• :

I i : : : ·········j···········j···········j···········j···········:

i i I I i ......... , ........... , ........... i···········i···········i

i i : : i

·······l···· ·1········l·······4········1

......... ! ......... 1 ........... i ........... i ........... 1

Factor: (x - 1)

co -, \ 0 _,

-, 0

_, 0 0 _,

-\"-)-

Root: 'X. � l � t> ! & _, �

--s 1 �

-� _,_ -.l a:.

rn -\ e

-

-.l -, {:i

Describe the behavior of the graph at this

Now, let's consider how we might be able to locate the zeros of a polynomial function numerically.

Consider the function x) = 2.x3 - .x2 ..:. 4x + 3 that we investigated earlier and whose graph is shownbelow. Find each pair of function values in the table below and answer the questions that follow

Find h(-2) and h(-1). Find h(O) and h(2).

��-ii-a� i:ij�jd!· ·······t·· · ·· ·· · ! · · ······ '!" • • • •• • • • !· · · · ··, ... l · ·.·.·.·. ·. ·.r,,. ·.·.·.·.·.·.· . ·.1,,. ·.· .·.·.·.·. ·. ·. r,,. ·. ·. ·.·.·.·.·.·.,,,.·· ·. ·.·.·.·.·.·.1,, . ·.·.·.·.· .·.·.·.1,,.!. !. !. !. :. ·. l. 33

- ...... = ........ = ........ : ........ = ........ = ........ =

From the graph, clearly h(x) has a zero between x = -2 and x = -1. Explain how your finding the valuesof h(-2) and h(-1) above numerically shows that there is a zero that exists between x = -2 and x = -1.

j S,"'C4. 'nl-�)=-C\ �O ,�j�d_\, �) \e,. loe.l��,e.-OJC.1s. R,t-:,J..

S,\l\tl.. h (- ,) : 41- > 01 -\"v..t. �� � t-\(;..) ', s Q...� � ,t-� \ � �:.- l .\,\(,-.) ,c, ��V\U� \o� X.� -_i �K.:.•I � � � �nv � 'X.-�\ -\-i, �� � x-�s .,3u�,t4&l� o.Does the same reasoning that you described concerning the zero between x = -2 and x = -1 hold true for °3e,Q the existence of a zero between x = 0 and x = 2? Explain your reasoning. -iJc.lc� �'\(A... hl1>):; >O � �ti)::.'l>O .� �T � �(r.) \\ � � � ?t-1>.>"i. � oJr. ?(.:. 0 o-J. X.: .2.. • � .. �

� � """"'�<L- � .f.M��c.c.. o(, °' �� lo� )(..�O � x::.� B� .. �,, --.�ocs,1'�� � Q.- �-t.

� ; 3QX"O w

'-'.ose �fa"

s 2- ...,.. l .

Based on what we have just seen, what inference can you make about the existence of a zero of a polynomial function if you know the value of the function at two different x - values?

G)-:t.� �(CL) � �('o) � �C\� ��� > 0.. F. � �fl..i.�� i. .., � J..., ��� .\,e � t,� �=� � 1(...:.-b.

©J;r �(,..) � �) � � ��. ���, �.r� �U.C.:. � .i.,•, fl<" 3 ; '- r-o�c 1blc � e.ASt' "� ,x.:..� ��:..b

The table of values below represents values of �function. The function has a negative root and three positive roots. Answer the questions that�

1. Is/ Are any of the roots of F(x) specifically identified in the table? What is the multiplicity of thiszero? Explain your reasoning for both questions

... l. • � .:. \ , � o.. � � �t�) 'o\C- � '>C:. \ 1 � c. 0 . SiYIC.L fi>.) \ ., .. "If om c. � , � s.11"" "b � M,\,1,#�'P L.: ... '\\�f\� �s � O.CAetJ.. �. � �-v� �s� �(L. i.�uJtl'\�· �04$. 1) �'"f�� :l. --rtvc;, � : ( Jw.J.J� � °'-. JW•-"-J�f Lc_�� � � Q.S. � j-\f� � V\»� � C \� O\'\ �� �iclL csf_ ?(_.:. l.

2. Between which-two x - .$1ahles in th�table is the negative root located? Ex�ain your reasoning.

� ��"" � � t=,.�) \ '> L.o�� 'ot.� ,x::.. -� � �-=-\ ¥>\c... F(-"f) -:: -40 > 0 � ft(-\).,:-� Z O �w..-� � � '� � �Vl>�� � � -�� c; lo� � :. - 4\- ONA- - \ .

3. Between which two x - values in the table is the second distinct positive root located? Explain your

�oni

�j 4l.�� �o�\�W (\.O� � F �) \ � l,oc.,.:l-t.el_b� �-:. � � ,c...:.S P\C. '1=ti) : -S:�O o.J'­'f"( 15) !: �b >O � � � j+ ���e\l"'t)� � 9l-� '> \o� 1t=- 3 � 'X-.=S.

Name �s�K� Date ���������-Period -��- Day #17 Homework

For exercises 1 - 3, study the graphs of the two functions that are given. Then, answer the questions that follow.

1. F(x)

Identify the zeros of the function and their multiplicities. ,

L )(-= - � > WV\U \�U"'J 3 'X-::. 3J �u\ti(>U�� 1

Based on the sum of the multiplicities, what type of function is F(x)?

Q \) G.V''t\ c__What are the guaranteed factors of F(x)?

(-x-+� � (-x-3)

2. G(x)

Identify the zeros of the function and their multiplicities. �=-'\-

) i(:l,�

...,_ ': �, o.Ll � 0..-

rt\ Ul 'hf� '4'. � i i. Based on the sum of the multiplicities, what type of function is G(x)?

Cubic_,

3. H(x)

Identify the zeros of the function and their multiplicities. •

?(=-1-) W\V\�Uc.r.!11� ')C::. I> MU l�U�f:j 1, 3

Based on the sum of the multiplicities, what type of function is H(x)?

Qv���C...

What are the guaranteed factors of What are the guaranteed factors of G(x)? H(x)?

(-x-t-�1(-x-i) o..vJ-6<-� �-+4) � c-x-DPictured to the right is the graph of a polynomial function, g(x). Use the graph to answer questions 4 - 8.

4. What type of function is g(x ). Give a reason for your answer. _ • " 4 r .1.1 'X ": .,. � 0.V\cl � � I \ �r� -fW �s 4"\. � ()c.) � __.. r-1-, H+-f-->++-rl

� <, °'" W\ u \hf'-'-�� � � h\c � 8�, � f------+--'f-i-+----+-_ -:_+-____.� � +o� ?<.-�s ��"�· �,�c,.),� �u .

5. Write an equation in factored form for tx). 6. Multiply the equation you wrote in exe�s� i,

�()() =. � -t_;>-)(x-ti:f.._x-0Cx-u the formg(x�= ax4

+hx3 +c;2

+dx+e.

j&-) -=- ( x.- � �--c?-) Cx -t � -;)J 3 3> ':l. ;1. jG<-) -=- J\ i -t x. -.?. �� -t x +_x -:lx-J)( -JJ:.·-\-1

°jC:I-J = XL\ +;;t)(.3 ·-.3X.:,1.-4xti

7. Based on the constant term of the equation you produced in exercise 6, why is this a reasonableequation for g(x) graphically?

� �Ci,� -1""(.N"-M. l�. � ��O't'\.) 4, �'4'�� 1.i.......�t'.e.-.> -ti,.. j-CH>� bf> -t-e j-i�c.4rt

J (D, 4),

8. Based on the graph of g(x), explain why we know that x = -2 is a zero of g(x) whose multiplicity is 2.Use synthetic division on the equation you wrote in exercise 6 to confirm this fact algebraically.

. � Explain why your work confirms this fact.� ?(.::. -J. � � IIW\U l.\-io�"- 4' �·, . 3 � '4 c!.J \ -i \ \ \) � \ ; -a - b -':L -i I � bl(: � j+\� ,

\-il- l o �3 _2 J.Q \ -"\ � � io -t£. -x:-a)(l� -A '1 -2 � x::.-�. Pr4ce-��. '- ��

\ _,_ I (ci � D i,.,4.1 �w-cl �CL t-, SJ��� 9. Using synthetic division, describe what the graph of h(x) = x4 - 6.x2- 8x - 3 looks like at the zero tltv, Sal·

of x = -1. Show your work and explain your reaso�. 8J I o - lo - S -3 \:!J l :�

-, ' · 5 � 8 El , -, -s -?> LA \ - . ,,

-1.,. th

�--c.. C. 0.. � ""4,--c{ 0 � �·ea,edl.8) \ -i -3 0 � �� ��-::-1 l�tl.��-,

[k """l�pu...;.� 3. �11.t""'- � -N �� -3 w�\.L �s � ,c.-a.)US � ?C:-1 � �°'-�

���-10. For what value of m is the value x = 2 a zero of the function.f{x) = -2.x3 + mx2 + x - 6?

tti -41 -"t

-;;,).

4�-20 -:-0

...:\-�:::. 2.0 ��s\

11. The value x = -2 is a zero of the function g(x) = x4 + 3x3 - 4x. Use synthetic division to determinethe multiplicity of the root. 00 I 3 D -4\

-2. -2. ':tr.:3\ \ \ -l. 0\;J -l. 2 O \-=-.-t \ -\ • 0 @..t:::!,l - 2.. -"'

0 0

?( = -;i \� Cl. � �se..

V'(\u\ � f \..\.C.: � ·, 5) �.

\ 2 8 12. The graph pictured to the right is the graph of g(x) = x4 + 3.x3-4x. Explain how

your work in exercise 11 is verified by the behavior of the graph at x = -2.

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-+u,-iu.

2 3

o � · \\� o. ��