Section 3.6 Complex Zeros
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1 04 Chapter Secton 6 Comple Zeros When fndng the zeros of polynomals, at some pont you're faced wth the problem Whle there are clearly no real numbers that are solutons to ths equaton, leavng thngs there has a certan feel of ncompleteness To address that, we wll need utlze the magnary unt, Imagnary Number The most basc comple number s, defned to be, commonly called an magnary number Any real multple of s also an magnary number Eample Smplfy 9 We can separate 9 as 9 We can tae the square root of 9, and wrte the square root of - as 9 = 9 A comple number s the sum of a real number and an magnary number Comple Number A comple number s a number z a b, where a and b are real numbers a s the real part of the comple number b s the magnary part of the comple number Arthmetc on Comple Numbers Before we dve nto the more complcated uses of comple numbers, let s mae sure we remember the basc arthmetc nvolved To add or subtract comple numbers, we smply add the le terms, combnng the real parts and combnng the magnary parts Eample Add 4 and 5 Addng 4) 5), we add the real parts and the magnary parts 4 5 5
2 6 Comple Zeros 05 Try t Now Subtract 5 from 4 We can also multply and dvde comple numbers Eample 4 Multply: 4 5) To multply the comple number by a real number, we smply dstrbute as we would when multplyng polynomals 4 5) = Eample 5 Dvde 5 ) 4 ) To dvde two comple numbers, we have to devse a way to wrte ths as a comple number wth a real part and an magnary part We start ths process by elmnatng the comple number n the denomnator To do ths, we multply the numerator and denomnator by a specal comple number so that the result n the denomnator s a real number The number we need to multply by s called the comple conjugate, n whch the sgn of the magnary part s changed Here, 4+ s the comple conjugate of 4 Of course, obeyng our algebrac rules, we must multply by 4+ on both the top and bottom 5 ) 4 ) 4 ) 4 ) To multply two comple numbers, we epand the product as we would wth polynomals the process commonly called FOIL frst outer nner last ) In the numerator: 5 )4 ) Epand Snce, ) Smplfy Followng the same process to multply the denomnator 4 )4 ) Epand
3 06 Chapter ) Snce, 6 )) =7 Combnng ths we get Try t Now Multply 4 and In the last eample, we used the conjugate of a comple number Comple Conjugate The conjugate of a comple number a b s the number a b The notaton commonly used for conjugaton s a bar: a b a b Comple Zeros Comple numbers allow us a way to wrte solutons to quadratc equatons that do not have real solutons Eample 6 Fnd the zeros of f ) 5 Usng the quadratc formula, ) 4)5) 6 4 ) Try t Now Fnd the zeros of f ) 4 Two thngs are mportant to note Frst, the zeros and are comple conjugates Ths wll always be the case when we fnd non-real zeros to a quadratc functon wth real coeffcents Second, we could wrte f ) 5 f we really wanted to, so the Factor and Remander Theorems hold
4 6 Comple Zeros 07 How do we now f a general polynomal has any comple zeros? We have seen eamples of polynomals wth no real zeros; can there be polynomals wth no zeros at all? The answer to that last queston, whch comes from the Fundamental Theorem of Algebra, s "No" Fundamental Theorem of Algebra If a non-constant polynomal f wth real or comple coeffcents wll have at least one real or comple zero Ths theorem s an eample of an "estence" theorem n mathematcs It guarantees the estence of at least one zero, but provdes no algorthm to use for fndng t Now suppose we have a polynomal f) of degree n The Fundamental Theorem of Algebra guarantees at least one zero z, then the Factor Theorem guarantees that f can be factored as f ) z q ), where the quotent q ) wll be of degree n- If ths functon s non-constant, than the Fundamental Theorem of Algebra apples to t, and we can fnd another zero Ths can be repeated n tmes Comple Factorzaton Theorem If f s a polynomal f wth real or comple coeffcents wth degree n, then f has eactly n real or comple zeros, countng multplctes If z, z,, z m, m,, m respectvely, are the dstnct zero of f wth multplctes f ) a z m m z z then m Eample Fnd all the real and comple zeros of f ) Usng the Ratonal Roots Theorem, the possble real ratonal roots are,,,,, 4 6 Testng, /
5 Chapter 08 Success! Because the graph bounces at ths ntercept, t s lely that ths zero has multplcty We can try synthetc dvson agan to test that The other real root appears to be or 4 Testng, Ecellent! So far, we have factored the polynomal to ) f We can use the quadratc formula to fnd the two remanng zeros by settng 0, whch are lely comple zeros ) 4)) ) The zeros of the functon are,,, We could wrte the functon fully factored as ) f When factorng a polynomal le we dd at the end of the last eample, we say that t s factored completely over the comple numbers, meanng t s mpossble to factor the polynomal any further usng comple numbers If we wanted to factor the functon over the real numbers, we would have stopped at ) f Snce the zeros of are nonreal, we call an rreducble quadratc meanng t s mpossble to brea t down any further usng real numbers It turns out that a polynomal wth real number coeffcents can be factored nto a product of lnear factors correspondng to the real zeros of the functon and rreducble quadratc factors whch gve the nonreal zeros of the functon Consequently, any nonreal zeros wll come n conjugate pars, so f z s a zero of the polynomal, so s z -/ /
6 6 Comple Zeros 09 Try t Now 4 Fnd the real and comple zeros of f ) Important Topcs of Ths Secton Comple numbers Imagnary numbers Try t Now Answers 4 ) 5 ) 9 4 ) ) 8 ) 4)4) ) Cauchy s Bound lmts us to the nterval [-, ] The ratonal roots theorem gves a lst of potental zeros:,, 5, 0 A quc graph shows that the lely ratonal root s = Verfyng ths, So f ) ) 5) Usng quadratc formula, we can fnd the comple roots from the rreducble quadratc ) ) 4)5) 6 4 ) The zeros of ths polynomal are,,
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