Sectin 3.2: Many f yu WILL need t watch the crrespnding vides fr this sectin n MyOpenMath! This sectin is primarily fcused n tls t aid us in finding rts/zers/ -intercepts f plynmials. Essentially, ur fcus turns t slving. As was mentined in the preceding lecture ntes, plynmials behave a lt like whle numbers. And just like with whle numbers, we can divide t bjects and get a qutient and remainder that are the same type f bject and are meaningful in the cntext f the divisin. We begin by lking back a ways all the way back t whle number lng divisin. We cnsider. The ld way: A new way: The ld way (r smething like it) is familiar t all f us. The new way needs sme explanatin. First, nte that and that S, the beginning stage f the lng divisin resembles the ld way (kind f expanded): In the ld way, we lk at hw many times 137 ges int 237. In the new way, we turn ur fcus t the leading terms, and. Hw many times des the first g int the secnd? Well, that s the same as the number f times 100 ges int 2000, which is 20. Hence the. 1
Nw, as in the ld way f lng divisin, yu multiply and subtract. Here, thugh, we need t multiply which explains the secnd rw under the divide symbl: Perfrming the subtractin and bringing dwn the 9 gives: Nw, we again fcus n the leading terms, and. Hw many times des the first g int the secnd? That s the same as hw many times des 100 g int -300, which is -3. This gives us the -3 that fllws the. Nw, multiplying the -3 with the gives us, which explains the secnd t last rw. Then we subtract and btain: Ntice that, and that, s that the answers are indeed the same either way. Furthermre, ntice that in the ld way we knw when t stp because the 50 was smaller 2
than the 137. In the new way, we knw when t stp because the leading term f the remainder is smaller than the leading term f the divisr. That is,. Nw, if we swap ut all the tens ( s) in the new way with s we will have perfrmed the plynmial lng divisin f Or rather, [Yu shuld try this nw. Cnfirm that yu get a qutient f, and a remainder f.] Nw, ne f the things which is different abut the plynmial lng divisin frm the new way f ding whle number lng divisin is that we can t get an intuitin r feel fr the quantities t the same extent. Fr example, when lking at hw many times went int, we culd ask urselves the mre basic questin, hw many times des 100 g int 2000? With the changever t s frm s, we get smething which lks like, hw many times des g int? The nature is identical, in the first case, in the secnd, but it feels significantly different. It takes practice t get cmfrtable with the new way, and plynmial lng divisin in general, and yu shuld watch vides nline, practice prblems nline where yu can get immediate right/wrng feedback, and practice prblems frm the textbk until yu are prficient and cmfrtable with the technique. A Hint: When trying t figure ut hw many times the leading term f the divisr ges int the leading term f the dividend, deal with the cefficients and the s separately. First, ask yurself what NUMBER times the leading cefficient f the divisr yields the leading cefficient f the dividend. THEN ask, hw many s frm the divisr g int the s f the dividend (multiplicatively which means yu ll need t knw yur expnent rules). When yu multiply yur guess by the leading term f the divisr, it MUST EQUAL the leading term f the dividend. If nt, yu re ding it wrng. - - - WOW, lk at all that terminlgy where d my nte card g again? Nw, just as in the whle number case, where we culd write, we can d the same with plynmials. THE EXACT SAME! S, in the whle number case, we have that and in the plynmial case, we have 3
Furthermre, lking at this in the way f mixed numbers, we have [ ] and, which cannt be made t lk analgus t the bracketed statement in the whle number case. Nw, ging back t the when d we stp? questin. We knw t stp plynmial lng divisin when the degree f the dividend is SMALLER than the degree f the divisr, at which pint the dividend becmes the remainder. This is because we can t fit any mre s int the qutient anymre. We can check ur answer by perfrming FOIL ing and simplifying and seeing if we get the dividend. In ur example, that wuld lk like [ ] Yu shuld cmpare the clr-cded line abve t all the parts f the plynmial lng divisin yu did fr this example. Nw, if we view the plynmial lng divisin with the dividend dented, the divisr dented, the qutient dented, and the remainder dented, we get the simple relatinship with the degree f guaranteed t be either less than the degree f, r the remainder will be 0. That is, either, r (the cnstant functin, zer fr any input ). Using the abve, if we take ur divisr t be, then we get which means that Thus, if we wish t evaluate the plynmial at, it s the same as dividing the plynmial by and plugging int the remainder plynmial,. [Remainder Therem] Yu might be wndering, why n earth wuld I want t g thrugh all the truble f plynmial lng divisin t evaluate a plynmial?! That s a gd questin! 4
That great questin leads us t the technique fr plynmial divisin knwn as synthetic divisin. Yur textbk walks thrugh a synthetic divisin example shwing ALL the glrius details (pages 259 & 260), and there are vides nline yu can watch, and lts and lts f examples fr yu t use fr practice. S I wn t bther ging ver the prcess s much, and I ll instead fcus n the interpretive aspects, and tips fr making gd use f the technique. First, taking the example in the textbk:, Sme cmments and tips (nte that my ntatin differs slightly frm the bk): The number in the upper left is the zer/rt/ -value making the divisr zer. The way students remember it is that it s the ppsite. S if the divisr was, the number in the upper left wuld be instead f. The number in the bttm right is the remainder! Using the remainder therem, this is the value f the plynmial being divided int, when yu plug in the number in the upper left crner fr, which means yu dn t need t bther with plugging in and then raising it t all thse pwers, and multiplying by all thse cefficients, and then adding and subtracting all that stuff! All that stuff is built int the technique! Building n this, in the example we have here, we see that the remainder is zer, which means that, which means is a rt/zer/ -intercept, and that is a factr f. S, this becmes a quick technique fr finding slutins t, as lng as we crrectly guess which numbers t put in the upper left crner. Ntice the preceding the. Since the dividend plynmial is degree three, these are the cefficients fr the qutient plynmial which has degree ne less. That is, the qutient is. This means that We have partially factred the plynmial! In fact, using ur extensive knwledge f quadratics, we knw that the qutient factrs as well, and we get the cmplete factrizatin: Nt particularly attractive, but there we have it ( ( )) ( ( )) 5
Additinally, t treat multiplicity, yu can iterate the synthetic divisin n the qutient, as in the example n page 263 (example 3.2.3). Once yu find a value f that wrks (i.e., is a zer/rt), yu iterate n subsequent qutients until the remainder isn t zer anymre, r until the qutient is quadratic. Once the qutient is quadratic, yu can use yur cnsiderable expertise with respect t quadratics t cmplete the factrizatin. Ntice that we culd try using 2 again n the abve, and we get: Since the 23 at the far right isn t 0, we knw that is a zer with multiplicity 1 (since the synthetic divisin nly gave a remainder f 0 t ne level f iteratin). We can d synthetic divisin with terrible, ugly lking numbers like t. But we try t avid it when pssible because it s errr prne and a little intimidating. It is really, really, imprtant t nte that we MUST include a 0 wherever a term is mitted in the riginal plynmial. That is, if we have a dividend f, we must treat it like, giving us the rw in the synthetic divisin. Yur answer is WRONG withut it!! AND this ges fr lng divisin t! It s s imprtant, I ll say it again - this becmes a quick technique fr finding slutins t, as lng as we crrectly guess which numbers t put in the upper left crner. which redirects ur fcus t making clever guesses fr values f that make the plynmial zer. Sme useful facts: The degree f a plynmial is the maximum number f -intercepts the plynmial can have, and is ftentimes mre than the number f -intercepts. S a 5 th degree plynmial abve has n mre than 5 -intercepts, and might even have less! 6
We abuse the language slightly and use the terms rts, zers, and -intercepts interchangeably. The BIG GRAY BOX n page 264 says the fllwing are equivalent: is a zer slves is a factr f is an -intercept Sectin 3.3: This sectin is fcused primarily n finding ways t make clever guesses fr the rts f a plynmial. Yur textbk ffers a useful, but nt entirely necessary therem: Cauchy s Bund Therem. This therem is helpful if yu knw hw t use it, but yu re safe in this class if yu ignre it. The Ratinal Rts Therem is anther useful therem, and yu ll be safe mst f the time if yu dn t knw hw t use it, but there will be a handful f exercises in which yu ll need t use it effectively. The Ratinal Rts Therem says that given a plynmial the pssible rts f the plynmial that can be written as a fractin are the ratis:, ALL f Fr example, taking the plynmial, we have, which has factrs, and, which has factrs. Taking all pssible ratis, we get Clearly, sme f these fractins are the same as thers, s if we remve them, we get the shrter list: Reducing these, we have all the pssible nice values fr that culd make the plynmial 0: S, we ve narrwed ur pssible number f guess frm but a TON better than infinitely many t guess! t 12. That s still mre than we d like, 7
[TIP: Take the path f least resistance! Start with the easy nes first, like. Then try the hard nes last (unless yu have sme suspicin that they ll wrk, then try them first). Als, we saw that we culd iterate the synthetic divisin n the qutient t find multiplicity. Well, after we ve gne as far as we can with ne f the zers in that way, we can take the last gd qutient (the last ne that had a remainder f 0), and ur next guess frm ur list f guesses t make, and d synthetic divisin frm there, withut having t start ver at the beginning. This will save us tns f time and sme aches in the head.] Cnsider the example:. The ratinal rts therem gives us the pssible guesses f. Perfrming the synthetic divisin fr 1 desn t wrk, but if we d -1, we get: S, -1 wrks twice, meaning that it is a zer f multiplicity 2. And the last gd qutient is the line that has the preceding the, which turns ut t be rather cnvenient because it s quadratic and we wn t have t try all the ther guesses generated abve by the ratinal rts therem. We have nw gtten t. Using ur knwledge f quadratics, we find that the zers f are. Thus, ( ) ( ) Ntice hw the 2 came ut frnt. We are actually making the int first, and then finding the zers f. Then we mve the 2 all the way ut frnt. [Make sure yu get LOTS & LOTS f practice with prblems like these!] 8