# Using the Rational Root Theorem to Find Real and Imaginary Roots Real roots can be one of two types: ra...-\; 0 or - l (- - ONLl --

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1 Using the Rational Root Theorem to Find Real and Imaginary Roots Real roots can be one of two types: ra...-\; 0 or - l (- - ONLl -- Consider the function h(x) =IJ\ 4-8x 3-12x x {?\whose graph is pictured below. What do you notice about this graph that does make sense based on the equation of the function? L,,, (D lta.,', ;-,, \,,,) -AO ki r. r,k, ""',. -rh Co-v\c;. -re...-- 'tr, ci,, s o.\.,o -\ '-'". 4 t ti\ =-1111\' c,.y.j..,\,s \Vhat c clusion can you draw about the four roots of the function,.f{x)? Explain your reasoning...ft- s J,s.-e o..ll 1)-k /\.co /\,,.Q_ - 0-MJ..... \ JQ.. I,)(_ The goal of this lesson is to use the Rational Root Theorem to aid us in finding all of the roots of the function whether they be rational, irrational, or imaginary. Make a list of the rational roots that the Rational Root Theorem guarantees are possible. t, 0 -,:i \ >!. J ± ct,\- \ _ 2,..,f'\,r\ - :. - ) -3,.., - I ± \, t 3 From this list, which roots appear to be roots from the graph above? Perform synthetic division to verify that they are, in fact, roots. Divide.f{x) by one of the associated factors. Then, divide that result by the other associated factor. en - -\ ':2. 2. 'i 1' -i ' x. = 3 U1 -\?) 3 -, Of P 'o?> -'\ - "1 [.Q. roo1"1> 1) V\U& Cl\ - I f\..co-k ><-') 7':. -\) 'J -J'3, ff.

2 Consider the function /(x) = 2x 4 - x 3 + 7x 2-4x-4 whose graph is pictured below :.....! :....!......, i : i : : t... ' + r i What do you notice about this graph that does not make sense based on the degree of the function?,1- ),s 4 'ou4- \ t'l) O -p -f 1 -P. -Ji What conclusion can you draw about the four roots of the ( " r T : - - " / ( r r ' < fw1 c t ion,j(x)? Explain your reasoning. : r r r :- -. : : : : i S,-rt..e.. roo SJ.S4. C1J\lZ. : r T r...,, j :, : n.u-l u.,kt ;1.. i::::::::::r:::::::r::::::1:::::::1::: :::r:::::::1:::::::1:::::::1:::::::::1 \)s t. Make a list of the rational roots that the Rational Root Theorem guarantees are possible.?.. -:. 1. \ ) t.2, t\\ +\ +.l "' ;-, t.1,, i:1.. - J - )--'l _., From this list, which roots appear to be roots from the graph above? Perform synthetic division to verify that they are, in fact, roots. Dividej(x) by one of the associated factors. Then, divide that result by the other associated fact9r. t.l \ 1 _ l '1 - 'i -'1 -= - i. 'f U>t"',.,. -, \ - m 'to Vl o..c... ol. ('Ot) - f> - g. m - g g (2: J5 lii (\,. You have just divided a degree 4 function twice so your tesuttmg polynomial 1s a C This function's roots will be the remaining two roots, which will be imaginary, ofj( Find these imaginary roots. 'X. 'l...-\" s ::. 0 AA = -S F i. = x-= :!. 1L

3 Consider the function f(x) = 12-(;;2( ; to answer the following questions. Make a chart of the possible comb ation f post'tive, n ative, zero, and imaginary roots forj(x). -t( ): \ ' '-\C\x)t a -t-:2.qx-\o N 'I Po. 1. '( 'if 'I do O 'i N. 3 OC'" i \ o o t o O Graphj(x) using a graphing calculator. Then, on the set of axes to the right, draw the graph. What conclusion can you make about the four roots of the function,f(x)? Explain your reasoning. _, _2-1 Si ss <; - i- "', po i-nut.. 0J..J.._ \Jt. t I -10 \-l -0- \;;,, " \e., s c.e-v' r c.j-. Make a list of the rational roots that the Rational Root Theorem guarantees are possible. 1'. ' R. -:, \ I j'. 2. J 1 c; I \ 0 :,.:t l J :t -:k I ± J t ).:t f '2. ) '± 2.) i!, i J :t\, :t2.i ;I :t J t(./t\, ::1: 5 :..S. ± :±:-5: ±.S:. ±10.:illt ;l) )... ) ) 12. ) J! ) From this list, which roots appear to be roots from the graph above? Perform synthetic division to verify that they are, in fact, roots. Dividej(x) by one of the associated factors. Then, divide that result by the other ajsociated factor. r=i, \ \ 1-& _ 2..-; - 10 =- \ -, -"t -\\ \0, :). I 'A \ Ip '-.. '4 'i -"to L.Q. 'Ji, <. '+ 't' -'to r You have just divided a degree 4 function twice so your resulting polynomial is a C..., -- 9;... t \, --r,;ie-, ' 2-'"t (oc) This function's roots will be the remaining two roots, which will be imaginary, of). Find these imagin roots. ')(. : _ b 1 J \o ':L_ 4,Q.C:.. t ><. -t x-+ <oo = 0,2.. l-::2.. 'I.. ').. ""T S 0

4 1. g(x) = 3x 3 + 8x 2 + l3x + 6 1'.R -:. :t \, :t 2., ± '>J ± 'P.R. -:. ±1, - 2:. 3 b ;t\' i ±.\,t, :t \, t3j ±-b 3 " "T bx. q "= 0 'L-t l.x.-\- : 0 X: -:t ± J4- (\)[i) - - im '2,(1).. ;l....,. -:..-± 't.j'i _ - \ ± i.ff- 2. h(x)=-6x 4 +x 3 +4x 2 +10x+3?.. ':. '± \ J '! ; P.R. :. \-L9. : -.)..!\ J t1, t3 J ±''- \ " " l ".l 3 + l -, -, - 3 > - c. ' - 'J - a \:1) -<c \ \0 2. -\ -\ m -b 3 3, -ta 3 '! 9 - -t.o -, - a -lo - ')C ,x - (. -: 0 '2,. "* \ ': [Q: 'I,. : - \ ':! j \ -1..\(.\)(.,) - -\ '± s=j c.,) ':lo x:.-,±.li3

5 Day #22 Homework Date Period --- For exercises 1-4, list the possible rational roots of the given function. Then, find all roots, real and imaginary, of the function. :t \0 ) t. if m -\ -! \ C\ I 2-,. 2,. -,;i. \ l -, :,. -s,o -10 ( uj (. \ \ -, -s Gt \1 lo,1 '5 u] <o \1 c; -2-5 (o 15 l..q (o -t,s-:.. 0 )(.:-\S "I..= 2. g(x)=6x., +19x -llx-14 ± \. ± l,! 1, t 1 "I + \ + 1 "'"l. _.;;.. -.; : - J -.))- 3),, t2, t 3/!:b \ 1'2 t 2., 12 tl Z t ) - I - J ) - J ;). ) J.) ±t,tt"t, ± \C\ lo 2S 2 l 2S - 2-\ -\-2\ 0 lo )I,. -:. - 1 \ \"\- _, t.\ ')(. : -,1,,_ 3. h(x) = 3x 4-8x 3 -l2x x + 9 :t\)! 3., t,_ +\!l tj! ' ±\, t.3.. -, 3J ) ' rn 3 -i - \ \\ ;11 3 ' -, -.3 LQ 3.:2- -q 0 3" ').. :: 4 "' =- 3 X=- J3 rn 3 \ I 0 t& 3 0-1

6 4. p(x) = 2x 3 +7x 2 +2x-3 ;l\ ) i?,.±..,, 1 +\ +l : - )-.:2.,-..,- \ - \ -3 3 x ')(..-\ -:=.O (".,.. - \ )( )(. + \ ) ::: \ -::..0 4-:., )(:. '/" The graph of a quintic - polynomial function, p(x), is shown to the right. Use the graph to answer questions If a is the leading coefficient of the equation of p(x), is a< 0 or is a > O? Give a reason for your answer. o1a f > \', Q:wl -\-1> s e, o.. o. 6. How many roots of p(x) are imaginary? Give a reason for your ; t)<.)'1') i... C...J o..q. s. ' _, - :: I l r """"s i s.1-+i.,..._\., _; r-oo s\. k.. / ii ) 4 3 '.!. :.r ; ' t 7. If c is the constant term of the equation of p(x), what is the value of c? Give a reason for your answer. \"' " -\ -\:.. st t.t. J '&-"f-\ \ lo,, c.-= Is it possible that there are four sign changes in the equation of p(x)? Give a reason for your answer. \)IM. ''t)"' l.d o..., ) N"t\A \.)M. u: 1 '"',;,,t. """ o..c...j. -w e&-,...6 \ MUW\. ) 0<,,11 \ue. \.t.. S \,)\ E>\'f. "l: ) 4c c; \ "'- )! k 4 1 oro

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