3.2 Properties of Division 3.3 Zeros of Polynomials 3.4 Complex and Rational Zeros of Polynomials

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1 Math Properties of Divisio 3.3 Zeros of Polyomials 3.4 Complex ad Ratioal Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered with fidig the solutios of polyomial equatios of ay degree that is, equatios of the form ( ) f x = ax + a x ax + ax+ a = 0 () 0 Defiitio A root or solutio of equatio () is a umber k that whe substituted for x leads to a true statemet. Thus, k is a root of equatio () provided f ( k ) = 0. We also refer to the umber k i this case as a zero of the fuctio f. Note that each real zero is a x-itercept of the graph of f. We are goig to fid aswers to the followig questios: a) How may zeros of f ( x) are real? Imagiary? b) How may real zeros of f ( x ) are positive? Negative? c) How may real zeros of f ( x ) are ratioal? Irratioal? d) Are the real zeros of f ( x ) large or small i value? Exercise # Checkig for a zero or root. a) Is - a zero of ( ) P x x x x b) Is x = a root of the equatio = + +? x 3x 0 + =? Note: If a root is repeated times, we call it a root of multiplicity 3 Exercise # a) State the multiplicity of each root of the equatio: x ( x ) ( x ) + = 0 b) Fid all zeros ad their multiplicities: f ( x) = 5x ( x+ )( x+ 5) c) Fid all zeros ad their multiplicities: ( ) ( ) ( ) f x = 7x x + 9

2 Divisio of Polyomials The process of log divisio for polyomials follows the same four-step cycle used i ordiary log divisio of umbers: divide, multiply, subtract, brig dow. Notice that i settig up the divisio, we write both the divided ad divisor i decreasig powers of x. Exercise #3 Divide by x +. 5x 6x 8x The result of the divisio ca be writte as: or Note ) Secod equatio is valid for all real umbers x, whereas first equatio carries implicit restrictios that x my ot equal -. For this reaso, we ofte prefer to write our results i the form of the secod equatio. ) The degree of the remaider is less tha the degree of the divisor. This is very similar to the situatio with ordiary divisio of positive itegers, where the remaider is always less tha the divisor. The Divisio Algorithm Let f ( x) ad p( x) be polyomials with p( x) of lower degree tha f ( x ) (3.) ad assume that p( x) 0. The there are uique polyomials q( x) ad r( x) such that f ( x) = p( x) q( x) + r( x) where r( x ) = 0 or the degree of r( x) is less tha the degree of ( ) The polyomials f ( x) ad p( x) are called the divided ad divisor, respectively, q( x) is the quotiet, ad r( x) is the remaider. p x. Whe r( x ) = 0, we have f ( x) = g( x) q( x) ad we say that g( x ) ad q( x ) are factors of ( ) f x. Exercise #4 Usig log divisio to fid a quotiet ad a remaider. Divide x + x 4 by x 3. Sythetic Divisio (3.) - Sythetic divisio is a quick method of dividig polyomials. - It ca be used whe the divisor is of the form x k. - I the sythetic divisio we write dow oly the essetial parts of the log divisio table (the coefficiets).

3 Exercise #5 (3. - #) Use sythetic divisio to perform the followig divisios: a) x 3x + 4x 5 x b) If f ( x) = x 3x + 4x 5, evaluate f ( ). What do you observe? 3 The Remaider Theorem Whe we divide a polyomial ( ) (3.) Proof f x by x c, the remaider is ( ) f c. Exercise #6 (3. - #) Usig the remaider theorem to evaluate a fuctio ad check for a factor. 4 f x x x x Let ( ) = i) Evaluate f ( 3). ii) Is x + 3 a factor of ( ) f x x x = ? The Factor Theorem The polyomial x c (3.) Proof is a factor of the polyomial f ( x) if ad oly if ( ) 0 f c =.

4 Exercise #7 Let ( ) f x x x x = 4 +. a) What is the remaider whe dividig the give polyomial by x? I how may ways ca you fid the remaider? b) Is x a factor of f ( x )? c) Is x a factor of f ( x )? 4 Exercise #8 Fid all values of k such that f ( x ) is divisible by the give liear polyomial. (3. - #39) f ( x) = kx + x + kx+ 3k + ; x + Exercise #9 Show that x c is ot a factor of ( ) (3. - # 4) 4 f ( x) = 3x + x + 5 f x for ay real umber c. Exercise #0 Let (, ) P x y be a first quadrat poit o y = 6 x ad cosider the vertical lie PQ with Q (3. - #47) o the x-axis. a) If PQ is rotated about the y-axis, determie the volume V of the resultig cylider. P x y with x is the volume V i part (a) the same as the volume of the cylider of radius ad altitude 5? b) For what poit (, ) The Fudametal Theorem of Algebra Every polyomial equatio of degree more tha or equal to ad complex coefficiets f ( x) = ax + a x ax + ax + a0= 0 (, a 0) has at least oe complex zero.(this zero may be a real umber). The Liear Factors Theorem (Complete Factorizatio Theorem for Polyomials) Every polyomial of degree > 0 ca be expressed as a product of liear factors. ( ) ( )( ) ( ) f x = a x x x x... x x, where a is the leadig coefficiet ad x i are the zeros of the polyomial. Theorem Every polyomial of degree has exactly roots, where a root of multiplicity k is couted k times.

5 Exercise # Write each polyomial as a product of liear factors. f x = 3x 5x a) ( ) f x x b) ( ) c) ( ) = 5 f x x x = Exercise # Fid the zeros of f ( x ), the express f ( ) (3.3 - #7, #) a) f ( x) = 4x + x + 9x 4 f x = x + x 00 b) ( ) x as a product of liear factors. Exercise #3 Factorig a polyomial give a zero. f x = 6x + 3x 4x+ 3. Show that -3 is a zero ad use this fact to factor a) Let ( ) f ( ) x completely. (3.3 #4) b) ( ) f ( ) f x x x x 4 = Kowig that 4 is a zero of multiplicity 4, factor x ito liear factors. Exercise #4 Fidig polyomial equatios satisfyig give coditios. (3.3 - #) Fid a polyomial fuctio of degree 3 havig the umbers -,, ad 3 as zeros ad satisfyig f ( ) = 80. Exercise #5 Fid a polyomial f ( ) x of degree 4 with leadig coefficiet such that both -5 ad are (3.3 - # 7) zeros of multiplicity, ad sketch the graph of f. Exercise #6 (3.3 - #3) Fid the polyomial fuctio of degree 3 whose graph is show i the figure

6 The Number ad Locatio of Real Zeros 6 Descartes Rule of Sigs I some cases, the followig rule discovered by the Frech philosopher ad mathematicia Ree Descartes aroud 637 is helpful i elimiatig cadidates from legthy lists of possible ratioal roots. x is a polyomial with real coefficiet, writte with descedig powers of x (ad omittig powers with coefficiet 0), the a variatio i sig is a chage from positive to egative or egative to positive i successive terms of the polyomial (adjacet coefficiets have opposite sigs). To describe this rule, we eed the cocept of variatio i sig. If f ( ) Example How may variatios i sig occur i the followig polyomial? ( ) f x = x x x + x + x Descartes Rule of Sigs Let f ( x ) be a polyomial with real coefficiets ad a ozero costat term. a) The umber of positive real zeros of f ( x) is either equal to the umber of variatios i sig i f ( x ) or is less tha that by a eve whole umber. b) The umber of egative real zeros of f ( x) is either equal to the umber of variatios i sig i f ( x) or is less tha that by a eve whole umber. Exercise #7 Use Descartes rule of sigs to determie the possible umber of positive real zeros ad (3.3 - #7, #3) egative real zeros for each fuctio, as well as the umber of oreal complex solutios. f x = x x + x a) ( ) b) ( ) f x = x + x x First Theorem o Bouds for Real Zeros of Polyomials Suppose that f ( x) is a polyomial with real coefficiets ad a positive leadig coefficie t ad that f ( x ) is divided sythetically by x c. ) If c > 0 ad if all umbers i the last row of the divisio process are either positive or f x. zero, the c is a upper boud for the real zeros of ( ) ) If c < 0 ad if the umbers i the last row of the divisio process are alterately positive ad egative ( ad a 0 is cosidered to be either positive or egative), the c is a lower boud for the real zeros of f ( x )

7 Exercise #8 Determie the smallest ad largest itegers that are upper ad lower bouds, respectively, for (3.3 - #36) the real solutios of the equatio. x 5x + 4x 8= 0 7 Whe a graphig utility is used, the followig theorem is helpful i fidig a viewig rectagle that shows all the zeros of a polyomial. Secod Theorem o Bouds for Real Zeros of Polyomials f x = ax + a x ax + ax+ a is a polyomial with eal coefficiets. Suppose ( ) 0 All of the real zeros of f ( x ) are i the iterval ( M, M), where M is the ratio of the largest coefficiet (i magitude) to the absolute value of the leadig coefficiet, plus. M ( a a a a0 ) max,,..., = + a Exercise #9 a) Fid a factored form for a polyomial fuctio (3.3 - #43) that has miimal degree. Assume that the itercept values are itegers. b) If the leadig coefficiet is, fid the y-itercept Exercise #0 A scietist has limited data o the temperature T (i degrees Celsius) durig a 4-hour period. (3.3 - #5) If t deotes time i hours ad t=0 correspods to midight, fid the fourth-degree polyomial that fits the iformatio i the followig table. t (hours) T ( C )

8 The Cojugate Zeros Theorem (3.4) If f ( x) is a polyomial fuctio with real coefficiets ad if a+ bi is a zero of f ( x ), the its cojugate a bi is also a zero of f ( x ). 8 Exercise # For the give polyomial, oe zero is give. Fid all the others. f x = x 7x + 7x 5 ; i ( ) Fidig all the ratioal zeros of a polyomial (3.4) The Factor Theorem tells us that fidig the zeros of a polyomial is really the same thig as factorig it ito liear factors. We ow study a method for fidig all the ratioal zeros of a polyomial. Example Cosider the polyomial f x = x x 3 x+ 4 Factored form ( ) ( )( )( ) = Expaded form. What are the zeros of f ( x )? What relatioship exists betwee the zeros ad the costat term of the polyomial? The ext theorem geeralizes this observatio. The Ratioal Zeros Theorem If the polyomial ( ) (3.4) ( a0 0, a 0 f x = ax + a x ax + ax+ a 0 ) has iteger coefficiets, the every ratioal zero of f ( x ) is of the form p q where p is a factor of the costat coefficiet a 0 q is a factor of the leadig coefficiet a. Note: The Ratioal Zeros Theorem gives oly POSSIBLE ratioal zeros. It does ot tell us whether these ratioal umbers are actual zeros.

9 Exercise # Usig the Ratioal Zeros Theorem Do each of the followig for the polyomial fuctio defied by 4 f x = 6x + 7x x 3x+. ( ) a) List all possible ratioal zeros. b) Fid all ratioal zeros ad factor f ( x ) ito liear factors. 9 Fidig the Ratioal Zeros of a Polyomial. List all possible ratioal zeros usig the Ratioal Zeros Theorem.. Use sythetic divisio to evaluate the polyomial at each of the cadidates for ratioal zeros that you foud i Step. whe the remaider is 0, ote the quotiet you have obtaied. 3. Repeat Steps ad for the quotiet. Stop whe you reach a quotiet that is a quadratic or factors easily, ad use the quadratic formula or factor to fid the remaiig zeros. Exercise #3 For the give polyomial fuctios, do the followig: a) ( ) (3.4 - #) b) ( ) i) List the maximum umber of real zeros; ii) List the umber of positive real zeros ad egative real zeros; iii) list all possible ratioal zeros; iv) fid all ratioal zeros; v) factor f ( x ). vi) Graph the fuctio. f x x x x = f x x x x x 5 4 = Exercise #4 A storage shelter is to be costructed i the shape of a cube with a triagular prism formig the (3.4 - #37) roof. The legth x of a side of the cube is yet to be determied. a) If the total height of the structure is 6 feet, fid a formula for the volume i terms of x. b) Determie x so that the volume is 80 cubic feet. Exercise #5 a) Fid all the complex zeros of ( ) 4 f x x x x x = b) Fid all the solutios of 4 x x x x = 0