Polynomial and Rational Functions. Polynomial functions and Their Graphs. Polynomial functions and Their Graphs. Examples

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1 Polomial ad Ratioal Fuctios Polomial fuctios ad Their Graphs Math 44 Precalculus Polomial ad Ratioal Fuctios Polomial Fuctios ad Their Graphs Polomial fuctios ad Their Graphs A Polomial of degree is a fuctio of the form P( ) a a a L a where is a oegative iteger ad a. The umber a a K a are coefficiets. a is the costat coefficiet or term. is the leadig coefficiet. a Eamples Eplai wh each of the followig fuctios is or is ot a polomial fuctio.. f() g(z) (z - ½). f() / 4. r().. / H ( ) 7 G( ) 7. Polomial Fuctios ad Their Graphs Polomial Fuctios ad Their Graphs 4

2 Graphs of Power Fuctios Graphs of Polomials ( ) ( ) 4 Polomial Fuctios ad Their Graphs Polomial Fuctios ad Their Graphs Ed Behavior of Polomials Ed behavior describes behavior (of ) becomes large i either positive or egative directio. me becomes large i the egative directio me becomes large i the positive directio For polomial with eve degree ad or ad For polomial with odd degree ad or ad Ed Behavior of Eve Degree Polomials Polomial Fuctios ad Their Graphs 7 Polomial Fuctios ad Their Graphs 8

3 Ed Behavior of Odd Degree Polomials Usig Zeros to Graph Polomials If P() is a polomial ad P(c) the a umber c is a zero of P. The followig statemets are equivalet c is a zero or root of P. c is a solutio of the equatio P(). c is a factor of P() Polomial Fuctios ad Their Graphs 9 Polomial Fuctios ad Their Graphs Eample Eample Fid the zeros for Fid the zeros for ( )( )( ) Polomial Fuctios ad Their Graphs Polomial Fuctios ad Their Graphs

4 Itermediate Value Theorem of Polomials If P is a polomial fuctio ad P(a) ad P(b) have opposite sigs the there eists at let oe value c betwee a ad b such that P(c). Problems o Page Sketch the graph of the fuctio.. P( ) ( )( )( ). P( ) a c f(b) b 7. P 4 ( ) f(a) Polomial Fuctios ad Their Graphs Polomial Fuctios ad Their Graphs 4 Local Etrema of Polomials Local Etrema of Polomials f() 8 If P() is a polomial degree the the graph of P h at most local etrema (turig poits). Polomial Fuctios ad Their Graphs Polomial Fuctios ad Their Graphs 4

5 Polomial Fuctios ad Their Graphs 7 Practice Problems o Page Dividig Polomial 8 Dividig Polomial Dividig Polomial 9 Log Divisio of Polomials Problem p. 78 The quotiet is ad the remaider is Dividig Polomial Divisio Algorithm If P() ad D() are polomials D() the there eist uique polomial Q() ad R() such that P() D() Q() R() where R() is either or of degree less tha the degree of D(). P() divided D() -- divisor Q() -- quotiet R() -- remaider

6 Sthetic Divisio Problem Page 7 Eample r7 Usig sthetic divisio to fid the quotiet ad the remaider of Dividig Polomial Dividig Polomial Remaider Theorem Factor Theorem If the polomial P() is divided b c the the remaider is the value P(c). Proof: If the divisor is i the form c for some real umber c the the remaider must be a costat (sice the degree of the remaider is less tha the degree of the divisor) P ( ) ( c) Q( ) r Where r is the remaider the P ( c) ( c c) Q( c) r r c is the zero of a polomial P() iff (if ad ol if) c is a factor of P(). Proof:. If c is a zero of P() that is P(c) the b the remaider theorem This implies c is a factor of P().. Let c be a factor of P() the Therefore P( ) ( c) Q( ) P( ) ( c) Q( ) P( c) ( c c) Q( c) Complete Factorizatio Theorem Dividig Polomial Dividig Polomial 4

7 Problems o Page 7 Fid a polomial of the specified degree that h the give zeros 7. Degree ; zeros: 8. Degree 4; zeros: 4 Practice Problems o Page Dividig Polomial Dividig Polomial Ratioal Zeros Theorem Real Zeros of Polomials The Polomial P( ) a a L a a h iteger coefficiets the ever ratioal zero of P is of the form p/q where p is a factor of the costat coefficiet a ad q is a factor of the leadig coefficiet a. Proof: If p/q is a ratioal zero i lowest terms the from Factor Theorem p p P a q q a p a p ( a p a p q L a q ) a q p a p q Sice p is the factor of the left had side p is also the factor of the right had side. Also sice p/q is i lowest terms p ad q have o factor i commo therefore p must be a factor of a. q L a pq p L a a q a q Real Zeros of Polomial 7 Real Zeros of Polomial 8 7

8 Problem 4 o Page 79 Problem o Page 79 List all possible ratioal zeros give b the Ratioal Zeros Theorem P( ) Possible ratioal zeros are ±/ ±/ ±/ ±4/ ±/ ±/ ±/ ±/ ±/ ±/ ±4/ ±/ ±/ ±/ ±/ ±4/ ±/ ±/ ±/ ±/ ±/ ±4/ ±/ ±/ ±/ ±/ ±/ ±4/ ±/ ±/ ±/ ±/ ±/ ±4/ ±/ ±/ Fid all ratioal zeros of the polomial.. P( ) Possible ratioal zeros are ±/ ±/ ±4/ ±8/ The zeros are ( ) ( 4) Real Zeros of Polomial 9 Real Zeros of Polomial Problem 8 o Page 8 Descartes Rule of Sigs Fid all ratioal zeros of the polomial. 4. P( ) 7 Possible ratioal zeros are ±/ ±/ ±/ ±/ ±/ 7 ( ) ( ) The zeros are / / Let P be a polomial with real coefficiets.. The umber of positive real zeros of P() is either equal to the umber of variatios i sig i P() or is less tha b a eve whole umber.. The umber of egative real zeros of P() is either equal to the umber of variatios i sig i P() or is less tha that b a eve whole umber. Real Zeros of Polomial Real Zeros of Polomial 8

9 Descartes Rule of sigs ad Upper ad Lower Bouds for Roots Polomial P() P() P() P() P() 4 P() 4 Variatios i sig Real Zeros or or Real Zeros or The Upper ad Lower bouds Theorem Let P be a polomial with real coefficiets.. If we divide P() b b (with b > ) usig sthetic divisio ad if the row that cotais the quotiet ad remaider h o egative etr the b is a upper boud for the real zeros of P.. If we divide P() b a (with a < ) usig sthetic divisio ad if the row that cotais the quotiet ad remaider h etries that are alteratel o-positive ad egative the a is a lower boud for the real zeros of P. Real Zeros of Polomial Real Zeros of Polomial 4 Problem 9 o Page 8 Problem o Page 8 Show that the give value for a ad b are lower ad upper boud for the real zeros of the polomial. 7. P( ) 8 9 9; a b Show that the give value for a ad b are lower ad upper boud for the real zeros of the polomial P( ) ; a b Real Zeros of Polomial Real Zeros of Polomial 9

10 Usig Algebra ad Graphig Devices to Solve Polomial Equatios Usig Graphig Devices to Solve Iequalities ( p. ) Real Zeros of Polomial 7 Real Zeros of Polomial 8 Usig Graphig Devices to Solve Iequalities ( p. ) Practice Problems o Page 79 4 > > Real Zeros of Polomial 9 Real Zeros of Polomial 4

11 Defiitio of Comple Numbers Comple Numbers A comple umber is a epressio of the form a bi where a ad b are real umber ad i. The real part is a. The imagiar part is b. Two comple umber are equal if ad ol if the real ad the imagiar part are equal. Comple Numbers 4 Comple Numbers 4. 4i. i Problems o Page 89 Arithmetic of Comple Numbers Additio: (a bi) (c di) (a c) (b d)i Subtractio: (a bi) (c di) (a c) (b d)i Multiplicatio: (a bi)(c di) (ac bd) (ad bc)i Divisio: a bi c di (a bi) (c di) (c di) (c di) (ac bd) (bc ad)i (c d) Comple Numbers 4 Comple Numbers 44

12 Square Root of Negative Numbers Powers of i If r is egative the the priciple square root of r is r i r The two square root of r are i r ad i r. i i i i i i 4 i i i If the remaider of / 4 4 the i the i the i the i i i i 7 i i 8 Comple Numbers 4 Comple Numbers 4 Problems o Page ( i) ( i ) 8. (4 i) ( i). ( i)( i) i 4. 4i Comple Roots of Quadratic fuctios i 8 Comple Numbers 47 Comple Numbers 48

13 Practice Problems o Page Comple Zeros ad the Fudametal Comple Numbers 49 Comple Zero ad the Fudametal Fudametal Complete Factorizatio Theorem Ever polomial P( ) a a L a a ( a ) with comple coefficiets h at let oe comple zero. If P() is a polomial of degree > the there eist comple umbers a c such that c K c (with a ) P( ) a( c)( c) L( c ) Proof: B the Fudametal P() h at let oe zero c. B the Factor Theorem P() ca be factored P( ) ( c ) Q( ) where Q () is of degree. Similarl Q () h at let oe zero c ad P( ) ( c) Q( ) where Q () is of degree. Cotiuig the process for steps we obtai a fial quotiet Q () is of degree a costat umber a. Therefore P ) a( c )( c ) L( c ) ( Zeros Theorem Comple Zero ad the Fudametal Comple Zero ad the Fudametal

14 Zeros ad Their Multiplicities Zeros Theorem Problems o page 98: Factor the polomial completel ad fid all its zeros. State multiplicit of each zero. 4. P( ) P( ) 4. P( ) 4 Ever polomial of degree h eactl zeros provided that a zero of multiplicit k is couted k times. Proof: Let P be a polomial of degree. B Complete Factorizatio Theorem P ) a( c )( c ) L( c ) ( Suppose c is a zero of P other tha c c K c P( c) a( c c )( c c) L( c c ) The c c c ad therefore P h eactl zeros. K c Comple Zero ad the Fudametal Comple Zero ad the Fudametal 4 Cojugate Zeros Theorem Problems o Page 98 If the polomial P h real coefficiets ad if the comple umber z is a zero of P the its comple cojugate z is also a zero of P. Proof: Let P( ) a a L a a where each coefficiet is real. Suppose that P(z). We must prove that P( z). P( z) a z a z a a z z L a z a L a z a Fid a polomial with iteger coefficiets that satisfies the give coditios.. Q h degree ad zeros i ad i 7. R h degree 4 ad zeros i ad with a zero of multiplicit. a z a z L a z a a z a z L a z a P( z) Comple Zero ad the Fudametal Comple Zero ad the Fudametal 4

15 Liear ad Quadratic Factors Theorem Ever polomial with real coefficiets ca be factored ito a product of liear ad irreducible quadratic factors with real coefficiets. Proof: If c a bi the ( c)( c) h real coefficiets. ( c)( c) [ ( a bi)][ ( a bi)] [( a) bi)][( a) bi)] ( a) ( bi) Problems o Page 99 Fid all solutios of the polomial equatio b factorig ad usig the quadratic formula. 48. P( ) P( ) a ( a b ) So if P is a polomial with real coefficiets the b the Complete Factorizatio Theorem P ) a( c )( c ) L( c ) ( Ad sice the comple roots occur i cojugate pairs the P ca be factored ito a product of liear ad irreducible quadratic factors with real coefficiets. Comple Zero ad the Fudametal 7 Comple Zero ad the Fudametal 8 Practice Problems o Page Ratioal Fuctios Comple Zero ad the Fudametal 9 Ratioal Fuctios

16 Ratioal Fuctios ad Asmptotes Graph of Ratioal Fuctio / A ratioal fuctio is a fuctio of the form P( ) r ( ) Q( ) Smbol a Meaig approaches a from the left P( ) r ( ) Q( ) a approaches a from the right goes to egative ifiit goes to positive ifiit Ratioal Fuctios Ratioal Fuctios Defiitio of Asmptotes Trformatios of /. The lie a is a vertical mptote of the fuctio f() if a or a a or a. The lie b is a horizotal mptote of the fuctio f() if b or The graph of ratioal fuctio of the form a b r( ) c d is the graph of shifted stretched ad or reflected. Ratioal Fuctios Ratioal Fuctios 4

17 Problems o Page Problems o Page 7. s ( ) if r( ) the s( ) r( ) 8. s ( ) if r( ) the s( ) r( ) Ratioal Fuctios Ratioal Fuctios Problems o Page Graphig Ratioal Fuctios. r( ) if s( ) the 9 r( ) r( ) 9s( ). Factor the umerator ad deomiator.. Fid - ad - itercepts. Fid vertical mptotes 4. Fid horizotal mptotes. Sketch the graph Ratioal Fuctios 7 Ratioal Fuctios 8 7

18 8 Ratioal Fuctios 9 Eample ) ( s itercept itercept mptote vertical ) ( mptote horizotal s Ratioal Fuctios 7 Problem o Page ) ( r itercept itercept ad ad mptote vertical ) ( mptote horizotal s Ratioal Fuctios 7 Problem o Page itercept o itercept ± itercept ad ad mptote vertical ) ( mptote horizotal s ) ( r Ratioal Fuctios 7 Asmptotes of Ratioal Fuctios Let r be the ratioal fuctio of the form. The vertical mptotes of r are the lies a where a is zero of the deomiator.. (a) if < m the r h horizotal mptote (b) if m the r h horizotal mptote (c) if > m the r h o horizotal mptote. ) ( b b b b a a a a r m m m m L L. m b a.

19 Practice Problems o Page b77bc8a. Ratioal Fuctios 7 9