ILLUSTRATING THE EXTENSION OF A SPECIAL PROPERTY OF CUBIC POLYNOMIALS TO NTH DEGREE POLYNOMIALS Dvid Miller West Virgini University P.O. BOX 6310 30 Armstrong Hll Morgntown, WV 6506 millerd@mth.wvu.edu Introdution Miller (011) disussed how to use the free intertive lger-geometry progrm Geoger, to investigte speil property of ui polynomils. The rtile demonstrted vrious wys to use Geoger to illustrte the speil property tht given ui polynomil with three rel zeros, tngent drwn to the urve t the point t whih the siss is the men of two of those three zeros will interset the horizontl xis t the other zero. This property ws extended y Miller nd Moseley (01) to higher degree polynomils. We will use Geoger to illustrte how this speil property of ui polynomils n e generlized to higher degree polynomils. This will e done y first reviewing the ui polynomil se through n pplet, disussing nd illustrting in Geoger the ses for fourth nd fifth degree polynomils long with derivtions of formuls. The proof for the 4 th degree polynomil se, nd the generl proof for the nth degree polynomil with zeros is shown in (Miller nd Moseley, 01). Cui Polynomils Miller (011) uses Geoger to present vriety of wys to illustrte different exmples of the speil property for ui polynomils. Insted of reiterting wht is everything desried out uis in Miller (011) nd Miller nd Moseley (01), we will onstrut n pplet in Geoger to illustrte the se where ll the zeros of the ui polynomil re rel numers nd then stte the proof. First go to www.geoger.org nd li on the downlod t tht ppers etween Aout nd Help ner the upper right orner. Downlod free version of Geoger y liing on the Applet Strt utton. Now follow the steps elow nd refer to Miller (011) under the suheding Adding More Visul Proof for more detiled instrutions. 1. Insert three sliders y liing on the slider menu nd liing in the grphing window three seprte times (see figure 1 elow). Lel the sliders,, nd with minimum -10, mximum 10, nd inrement of 0.01. Figure 1: Slider Button. In the input ox (ottom of Geoger) type in 153
f(x) = (x-)*(x-)*(x-). Your sreen should loo lie figure elow. Figure : Generl Cui with ll zeros t 1. 3. Adjust to e, to e 1, nd to e -1. Your sreen should loo similr to Figure 3 elow. Figure 3: Cui with Speified Sliders 4. Now we need to pture the x-interepts. Strt typing in roots nd when the input ox gives you drop down menu of different ommnds with root in them, selet the ommnd Roots[ <Funtion>, <Strt x-vlue>, <End x-vlue> ] nd insert f(x) for <Funtion>, -10 for <Strt x-vlue>, 10 for <End x-vlue>, nd Enter. You should get points A, B, nd C for the x-interepts of the polynomil. Your sreen should loo lie figure 4 elow. 5. Input into the input r, g(x) = Derivtive[f(x)]. Cli on the rdio utton next to g(x) in the lger ox (under the dependent ojets) to hide it. 6. Type in d = ()/ into the input ox. Repet y typing e=()/ nd h=()/. 154
Figure 4: Cui with Mred Zeros 7. Now type in to the input ox seprtely, y f(d)=g(d)*(x-d), y f(e)=g(e)*(x-e), nd y f(h)=g(h)*(x-h). Your sreen should loo similr to Figure 5 elow. Figure 5: Cui with Tngent Lines 8. Polish up the sreen y inserting olor, line thiness, y doing the following: Right li on f(x) nd go to ojet properties. Choose olor (I hoose lue) nd line thiness (under style) to e 4. Repet for eh tngent line nd the sliders. Under ojet properties si, you n lel f(x) nd eh tngent line y seleting the ox for show lel nd liing on the down rrow in the seletion ox nd hoosing nme & vlue option. Your sreen should loo similr figure 6 elow. 155
Figure 6: Polished Version of the Cui Polynomil with Sliders Notie tht s you move the sliders to different vlues, the tngent line t the verge of two of the zeros lwys hs n x-interept t the other zero. This shows ll the different exmples for rel zeros (even in the trivil se when two or three of the zeros re the sme), exept when two of the zeros re omplex onjugtes. The proof elow shows why this lwys wors for ui polynomils with zeros,, nd nd n e found in (Miller nd Moseley, 01). Students elieve tht they n sy something is true (proved) if they show some exmples. Therefore it might e est to emphsize why the proof is neessry. Given ny ui polynomil, we n ftor it y the fundmentl theorem of lger into the form ) )( )( ( ) ( x x x x f = where,, nd re omplex numers (notie in the Geoger pplet, = 1. One ould insert nother slider to inorporte the sling ftor ). Assume tht. Otherwise the derivtion is trivil when = or = =. Evluting the funtion t the verge of nd, we hve = = f. Evluting the derivtive t the verge of nd, we hve = = f So the tngent line is = x y Sustituting y = 0 nd solving for x we hve 156
0= x 0= 0= x x This lst eqution simplifies to 0 = x nd so x=, the other zero. Sine there ws no preferene why we pied the zeros nd to verge, we would otin the sme result if we hose ny two distint zeros. We see n wesome property of ui polynomils, ut we re left with feeling tht there is more to this tht wht meets the eye. Is there some hidden explntion for why it wors? Does the tngent line to n nth degree polynomil t the verge of ll ut one of the zeros lwys hve n x-interept t the other zero, for? If not, is there nother formul in terms of ll ut one of the zeros, esides verging, tht would yield the sme result? Although we do not nswer these questions expliitly in this rtile, the reder n referene (Miller nd Moseley, 01) to get more detils of the formuls, underlying property from Clulus, nd generl proofs. We will onentrte on uilding Geoger pplets with sliders to illustrte how the ui property n e generlized to higher degree polynomils nd use the formuls in the pplets. Fourth Degree Polynomil Exmple Now we will loo t uilding exmples of fourth nd fifth degree polynomils. Let,,, nd d e the rel zeros of the fourth degree polynomils (zeros ould e omplex, ut n not e visulized in Geoger). The formul for fourth degree polynomils f(x) is derived in (Miller nd Moseley, 01). Given three of the zeros of f(x), sy,, nd, nd the solutions, nd, of the following qudrti eqution 3x ( ) x = 0, the tngent line t nd, intersets the x-xis t the other zero d. To illustrte this we will uild Geoger pplet with the following steps. 1. Insert four sliders y liing on the slider menu nd leling them,,, nd d with minimum -10, mxium 10, nd inrement of 0.01.. In the input ox (ottom of Geoger) type in f(x) = (x-)*(x-)*(x-)*(x-d). Your sreen should loo figure 7 elow. 3. Adjust to e -, to e -1, to e 0, nd d to e 1. Your sreen should loo lie figure 8 elow. 157
4. Now we need to pture the x-interepts. Strt typing in roots nd when the input ox gives you drop down menu of different ommnds with root in them, selet the ommnd Roots[ <Funtion>, <Strt x-vlue>, <End x-vlue> ] nd insert f(x) for <Funtion>, -10 for <Strt x-vlue>, 10 for <End x-vlue>, nd Enter. You should get points A, B, C, nd D for the x-interepts of the polynomil. Your sreen should loo lie figure 9 elow. 5. Input into the input r, g(x) = Derivtive[f(x)]. Cli on rdio utton next to g(x) in the lger ox (under the dependent ojets) to hide it. 6. Type in e=(sqrt(^^^-()))/3 nd h=(-sqrt(^^^-()))/3 into the input ox. 7. Now type in y-f(e)=g(e)*(x-e) to get the tngent line t e tht hs n x-interept t d. Repet for tngent line t h y inputing y-f(h))=g(h)*(x-h).your sreen should loo lie figure 10 elow. 8. Polish up the sreen y inserting olor, nd line thiness (see figure 11). FIGURE 7: Generlized 4 th degree polynomil with sliders set t 1 FIGURE 8: Generlized 4 th degree polynomil with sliders for the zeros. FIGURE 9: Generlized 4 th degree with zeros leled A,B, C, nd D 158
FIGURE 10: Generlized 4 th degree polynomil w/ tngent lines tht interset t D FIGURE 11: Polished Up Version of 4 th degree polynomil It is to note tht we re only showing the tngent lines tht interset t d. There re six more tngents lines, two tht hve x-interepts t, two t, nd two t. The x-vlues of the point of tngeny n e lulted y formul similr to the formul in 6 ove (the formul would e in terms of,,d, nd e for the tngent lines tht interset the horizontl xis t ). If you would onstrut ll the tngent lines then you would get the follow view iin Geoger (see figure 1). Fifth Degree Polynomil Exmple Let,,, d, nd e e the rel zeros of the fifth degree polynomil f(x). The formul for f(x) is derived in (Miller nd Moseley, 01). Given four of the zeros of f(x), sy,,, nd d, the solutions of the ui eqution sy,, nd the tngent line t nd, nd intersets the x-xis t the other zero e. To illustrte this we will uild Geoger pplet with the following steps. 1. Input five sliders nd lel them,,, d, nd e. Selet Min to e -10, Mx to e 10, nd Inrement to e 0.001.. Input into the Input ox f(x) = (x-)*(x-)*(x-)*(x-d)*(x-e) nd slide the sliders so tht is -, is -1, is 0, d is 1, nd e is. 3. Type in Roots[ f(x), -10, 10 ]. Your sreen should loo similr to figure 13 elow. 159
4. Now we wnt to find the solutions to the (ui) eqution we get from the proof for higher degree polynomils. In (Miller nd Moseley, 01) we see tht for higher degree polynomils we hve n 1 n1 i= 1 j= 1 i j ( x0 x j ) = 0 where nd re four of the five zeros of the fifth degree polynomil. Expnding this out we hve Let nd (prmeters orresponding to the prmeters in the pplet). Figure 1: Qurti showing ll tngent lines. FIGURE 13: 5 th degree polynomil with sliders To solve this we will type in the input ox the two ommnds nd Roots[ m(x), -10, 10 ]. 160
The roots re the three points where the tngent line hs n x-interept t the other zero e. Your sreen should loo similr to figure 14 elow (where the line type (under ojet properties tht you n get to y right liing on m(x)) ws hnged to dotted). 5. To find the zeros of m(x), type in Root[m(x)].You should see the points F, G, nd H pper on the grph nd in the lger ox. 6. These re the three points where the tngent line (t these points) hve x-interepts t e. To see this we first must otin the x-oordintes of F, G, nd H y typing x(f), x(g), nd x(h). These will pper in the lger window s g, h, nd I (sine f ws used for the funtion). 7. Type j(x)=derivtive[f(x)] nd hide it. Figure 14: 5 th degree polynomil with dotted derivtive 8. Now type y-f(g)=j(g)*(x-g), y-f(h)=j(h)*(x-h), nd y-f(i)=j(i)*(x-i). 9. Now polish things up y dding olor, line thiness nd leling. Your sreen should e similr to figure 15. FIGURE 15: Polished up Version 5 th degree polynomil 161
Conlusion This rtile hs shown how to use Geoger to extend mysterious property of ui polynomils to more generl properties for fourth nd fifth degree polynomils. In generl one n illustrte this for n nth degree polynomil to find the n- points on the x-xis in terms of n-1 zeros, tht is given y the solutions to the eqution elow, where re the n-1 zeros, n 1 n1 i= 1 j= 1 i j ( x x j ) = 0 suh tht the tngent line to the nth degree polynomil t eh intersets the x-xis t the other zero. Here for re the zeros of the resulting n- degree polynomil derived from the formul ove. See (Miller nd Moseley, 01) for more detils nd the underlying lulus onept ehind the generl property for polynomils. The reder should illustrte this for the 6 th degree polynomil nd thin out the proof efore referening the rtiles. This rtile shows some si funtions of Geoger so tht the reder n fmilirize themselves with the progrm nd is good tool for students to disover some mthemtis out polynomils in whih they n see some speifi exmples nd wor on more generl proof vi pper nd penil. Referenes Hmilton, Pete. (005). Mthemtis IB Higher Portfolio: Zeros of Cui Funtions. http://www.srid.om/do/18118175/zeros-of-cui-funtions (essed 11-8-011). Miller, D. (011). Investigting Zeros of Cuis through GeoGer. Mthemtis Teher, Vol. 105, No.. Miller, D., nd Moseley, J. (01). Extending Property of Cui Polynomils to Higher-Degree Polynomils. MthAMATYC Edutor, Vol. 3, Issue 3. Stewrt, J. (007). Essentil Clulus: Erly Trnsendentls. Thomson Broos/Cole. Sullivn nd Sullivn. (006). Alger nd Trigonometry: Enhned with Grphing Utilities. Person Prentie Hll, 4 th Edition. 16