Unit 4: Polynomial and Rational Functions

Transcription

1 48 Uit 4: Polyomial ad Ratioal Fuctios Polyomial Fuctios A polyomial fuctio y px ( ) is a fuctio of the form p( x) ax + a x + a x ax + ax+ a where a, a 1,..., a, a1, a0are real costats ad are called the coefficiets of px ( ). is the degree of px ( ) ad is a positive iteger. a is called the leadig coefficiet ad a0 is the costat term of the polyomial. The domai of ay polyomial is all real umbers. Ex. 1 Determie the degree, the leadig coefficiet ad the costat term of the polyomial. a) 4 5x + 7x x+ 7 b) gx ( ) 1x + 5x 4x Ed Behavior of a Polyomial There are four scearios: 1) Sketch px ( ) x, px ( ) x ( is eve, a > 0 ) 4 ) Sketch px ( ) x, px ( ) x ( is eve, a < 0 ) 4 As x, px ( ) As x, px ( ) As x, px ( ) As x, px ( )

2 49 ) Sketch px ( ) x, px ( ) x ( is odd, a > 0 ) 5 4) Sketch px ( ) x, px ( ) x ( is odd, a < 0 ) 5 As x, px ( ) x, px ( ) As x, px ( ) x, px ( ) As x ad x, the graph of the polyomial p( x) ax + a x + a x ax + ax+ a resembles the graph of y ax. Ex. Use the zeros ad the ed behavior of the polyomial to sketch a approximatio of the graph of the fuctio. a) x 9x b) gx x x 4 ( ) 5 + 4

3 50 c) + 5 x x Repeated Zeros If a polyomial has a factor of the form ( x c) k, where k > 1, the x cis a repeated zero of multiplicity k. If k is eve, the graph of flattes ad just touches the x -axis at x c. If k is odd, the graph of flattes ad crosses the x -axis at x c. Ex. 4: Sketch the give graphs x x + x 4 gx x x+ x ( ) ( 1) ( )( )

4 51 Ex. 5: The cubic polyomial px ( ) has a zero of multiplicity two at x 1, a zero of multiplicity oe at x, ad p( 1). Determie px ( ) ad sketch the graph. Ex. 6: A ope box is to be made from a rectagular piece of cardboard that is 1 by 6 feet by cuttig out squares of side legth x feet from each corer ad foldig up the sides. a) Express the volume of the box vx ( ) as a fuctio of the size x cut out at each corer. b) Use your calculator to approximate the value of x which will maximize the volume of the box. Ex. 7: The differece of two o-egative umbers is 10. What is the maximum of the product of the square of the first umber ad the other?

5 5 The Itermediate Value Theorem Suppose that f is cotiuous o the closed iterval [ abad, ] let N be ay umber betwee f( a) ad f( b ), where f( a) f( b). The there exists a umber c i ( ab, ) such that f() c N. Ex. 1: Verify that the Itermediate Value Theorem applies to the idicated iterval ad fid the value of c guarateed by the theorem. x + x 1, [0,5], f( c ) 11 Ex. : Show that there is a root of the equatio x + x 1 0i the iterval (0,1).

6 5 The Divisio Algorithm Let ad d( x) 0be polyomials where the degree of is greater tha or equal to the degree of d( x ). The there exists uique polyomials qx ( ) ad rx ( ) such that rx ( ) qx ( ) + or d( xqx ) ( ) + rx ( ). d( x) d( x) where rx ( ) has a degree less tha the degree of d( x ). Ex. 1: Divide the give polyomials. a) 6x 19x + 16x 4 x b) x 1 x 1 c) x x x+ 6 x + 1

7 54 Remaider Theorem If a polyomial is divided by a liear polyomial x c, the the remaider r is the value of at x c. I other words, f() c r Ex. : Use the Remaider Theorem to fid r whe 4x x + 4is divided by x. Ex. : Use the Remaider Theorem to fid f() c for 4 x 5x + 7 whe 1 c Sythetic Divisio Sythetic divisio is a shorthad method of dividig a polyomial px ( ) by a liear polyomial x c. It uses oly the coefficiets of px ( ) ad must iclude all 0 coefficiets of px ( ) as well. Ex. 4: Use sythetic divisio to fid the quotiet ad remaider whe a) f x ( ) x 1 is divided by x 1 b) 4 x 14x + 5x 9 is divided by x + 4 c) x x + x + x is divided by x 4

8 55 Ex. 5: Use sythetic divisio ad the Remaider Theorem to fid f() c for x + 4x + x 8x 6x + 9 whe c. Ex. 6: Use sythetic divisio ad the Remaider Theorem to fid f() c for x 7x + 1x 15 whe c 5. The Factor Theorem A umber c is a zero of a polyomial px ( ) ( pc () 0) if ad oly if ( x c) is a factor of px ( ). Examples: Determie whether a) x + 1is a factor of f x x x + x 4 ( ) b) x is a factor of x x + 4

9 56 Fudametal Theorem of Algebra A polyomial fuctio px ( ) of degree > 0 has at least oe zero. I fact, every polyomial fuctio px ( ) of degree > 0 has at exactly zeros. Complete Factorizatio Theorem Let c1, c,... c be the (ot ecessary distict) zeros of the polyomial fuctio of degree > 0 : p( x) ax + a x + a x ax + ax+ a The px ( ) ca be writte as the product of liear factors p( x) a ( x c )( x c ) ( x c ). 1 Ex. 1: Give the complete factorizatio of the give polyomial px ( ) with give iformatio: a) px ( ) x 9x + 6x 1; 1 x is a zero. b) px x x x + x+ ; x, x 5 are both zeros. 4 ( )

10 57 c) px ( ) x 6x 16x+ 48 ; ( x ) is a factor. d) + ; x(x 1) is a factor. 4 px ( ) x 7x 5x x Ex. : Fid a polyomial fuctio of degree three, with zeros 1,-4, 5 such that the graph possesses the y - itercept (0,5).

11 58 The Ratioal Zero Test Suppose p q is a ratioal zero of 1 ( ) f x ax a x a x ax + ax+ a, where a0, a1..., a are itegers ad a 0. The p divides a0 ad q divides a. The Ratioal Zero Test provides a list of possible ratioal zeros. Examples: Fid all the ratioal zeros of the factor the polyomial completely. a) f x x x x x 4 ( ) b) f x x x x x 4 ( ) + +

12 59 Complex Roots of Polyomials Cosider factorig the fuctio: f x ( ) x 1 The Square Root of -1 We defie i 1 so that i 1. Complex Numbers A complex umber is a umber of the form a + bi where a ad b are real umbers. The umber a is called the real part ad the umber b is called the imagiary part. Complex Arithmetic Ex. 1: a) ( + i) (6 i) b) ( + i)(4 i) c) ( 6 i)( + 6 i) d) (4 5 i)(4 + 5 i) Complex Cojugates The complex cojugate for a complex umber z a + bi is z a bi. I geeral, ( a bi)( a + bi)

13 60 Ex. : Simplify. a) ( + i) (1 6 i) ( i) (1+ 7 i) Ex. : Simplify. a) 4 b) 8 Ex. 4: Determie all solutios to the equatio x 4x+ 1 0 Ex. 5: Completely factor f x ( ) x 1.

14 Ex. 6: Fid the complete factorizatio of multiplicity two x 1x + 47x 6x+ 6 give that 1 is a zero of Cojugate Pairs of Zeros of Real Polyomials If the complex umber z a + bi is a zero of some polyomial px ( ) with real coefficiets, the its cojugate z a bi is also a zero of px ( ). Ex. 7: Fid a rd degree polyomial gx ( ) with real coefficiets ad a leadig coefficiet of 1 with zeros 1 ad 1 i. 4 Ex. 8: 1+ i is a zero of x x 4x + 18x 45. Fid all other zeros ad the give the complete factorizatio of f( x ).

15 6 Ratioal Fuctios A ratioal fuctio y is a fuctio of the form fuctios. px ( ), where p ad q are polyomial qx ( ) Ex. 1: Recall the paret fuctio 1. Use trasformatios to sketch x gx ( ) x 1 Asymptotes of Ratioal Fuctios The lie x ais a vertical asymptote of the graph of if or as x a + (from the right) or x a (from the left). Vertical Asymptotes px ( ) The graph of has vertical asymptotes at the zeros of qx ( ) after all of the commo factors qx ( ) of px ( ) ad qx ( ) have bee caceled out; the values of x where qx ( ) 0ad px ( ) 0. Holes The graph of px ( ) has a hole at the values of x where qx ( ) 0ad px ( ) 0. qx ( )

16 6 Horizotal Asymptotes The lie y bis a horizotal asymptote of the graph of if bwhe x or x. I particular, with a ratioal fuctio px ( ) ax + a x ax+ a m q( x) b x + b x bx+ b m m 1 m There are three cases: 1. If < m, the y 0 is the horizotal asymptote. x Ex: 1x + 7x a. If m, the y is the horizotal asymptote. b m Ex: x + 6x 4x + x. If > m, the there is o horizotal asymptote. Ex: 4 x x 5x x x Slat Asymptote If the degree of umerator is exactly oe more tha the degree of the deomiator, the graph of has a slat asymptote of the form y mx + b. The slat asymptote is the liear quotiet foud by dividig px ( ) by qx ( ) ad essetially disregardig the remaider. Example: x 8x+ 1 x + 1

17 64 Ex. : Fid all asymptotes ad itercepts ad sketch the graphs of the give ratioal fuctios: a) x 1 Domai: Rage: Equatio(s) of vertical asymptotes: Equatio(s) of horizotal asymptotes: Equatio of slat asymptote: x - itercepts: y - itercept: b) x + x + 4 Domai: Rage: Equatio(s) of vertical asymptotes: Equatio(s) of horizotal asymptotes: Equatio of slat asymptote: x - itercepts: y - itercept:

18 65 x + c) ( x )( x+ 5) Domai: Rage: Equatio(s) of vertical asymptotes: Equatio(s) of horizotal asymptotes: Equatio of slat asymptote: x - itercepts: y - itercept: d) x x x + 1 Domai: Rage: Equatio(s) of vertical asymptotes: Equatio(s) of horizotal asymptotes: Equatio of slat asymptote: x - itercepts: y - itercept: x + 1 e) x x Domai: Rage: Equatio(s) of vertical asymptotes: Equatio(s) of horizotal asymptotes: Equatio of slat asymptote: x - itercepts: y - itercept:

19 66 f) (x+ 1)( x ) ( x )( x+ 1) Domai: Rage: Equatio(s) of vertical asymptotes: Equatio(s) of horizotal asymptotes: Equatio of slat asymptote: x - itercepts: y - itercept: Ex. : Sketch the graph of a ratioal fuctio that satisfies all of the followig coditios: + as x 1 ad as x 1 as x ad as x has a horizotal asymptote y 0 has o x -itercepts Has a local maximum at ( 1, ) + Ex. 4: The product of two o-egative umbers is 60. What is the miimum sum of the two umbers?