Polynomial and Rational Functions

Polnomial and Rational Functions Figure -mm film, once the standard for capturing photographic images, has been made largel obsolete b digital photograph. (credit film : modification of ork b Horia Varlan; credit memor cards : modification of ork b Paul Hudson) CHAPTER OUTLINE. Quadratic Functions. Poer Functions and Polnomial Functions. Graphs of Polnomial Functions. Dividing Polnomials. Zeros of Polnomial Functions. Rational Functions.7 Inverses and Radical Functions.8 Modeling Using Variation Introduction Digital photograph has dramaticall changed the nature of photograph. No longer is an image etched in the emulsion on a roll of film. Instead, nearl ever aspect of recording and manipulating images is no governed b mathematics. An image becomes a series of numbers, representing the characteristics of light striking an image sensor. When e open an image file, softare on a camera or computer interprets the numbers and converts them to a visual image. Photo editing softare uses comple polnomials to transform images, alloing us to manipulate the image in order to crop details, change the color palette, and add special effects. Inverse functions make it possible to convert from one file format to another. In this chapter, e ill learn about these concepts and discover ho mathematics can be used in such applications. Donload for free at http://cn.org/contents/9b08c9-07f-0-9f8-dad9970d@.9

SECTION. GRAPHS OF POLYNOMIAL FUNCTIONS 7 LEARNING OBJECTIVES In this section, ou ill: Recognize characteristics of graphs of polnomial functions. Use factoring to find zeros of polnomial functions. Identif zeros and their multiplicities. Determine end behavior. Understand the relationship beteen degree and turning points. Graph polnomial functions. Use the Intermediate Value Theorem.. GRAPHS OF POLYNOMIAL FUNCTIONS The revenue in millions of dollars for a fictional cable compan from 00 through 0 is shon in Table. Year 00 007 008 009 00 0 0 0 Revenues..8. 9. 8. 8. 8.7 7. The revenue can be modeled b the polnomial function Table R(t) = 0.07t +.t 9.777t + 8.9t 0. here R represents the revenue in millions of dollars and t represents the ear, ith t = corresponding to 00. Over hich intervals is the revenue for the compan increasing? Over hich intervals is the revenue for the compan decreasing? These questions, along ith man others, can be ansered b eamining the graph of the polnomial function. We have alread eplored the local behavior of quadratics, a special case of polnomials. In this section e ill eplore the local behavior of polnomials in general. Recognizing Characteristics of Graphs of Polnomial Functions Polnomial functions of degree or more have graphs that do not have sharp corners; recall that these tpes of graphs are called smooth curves. Polnomial functions also displa graphs that have no breaks. Curves ith no breaks are called continuous. Figure shos a graph that represents a polnomial function and a graph that represents a function that is not a polnomial. f f Figure Donload for free at http://cn.org/contents/9b08c9-07f-0-9f8-dad9970d@.9

7 CHAPTER POLYNOMIAL AND RATIONAL FUNCTIONS Eample Recognizing Polnomial Functions Which of the graphs in Figure represents a polnomial function? g f h k Solution Figure The graphs of f and h are graphs of polnomial functions. The are smooth and continuous. The graphs of g and k are graphs of functions that are not polnomials. The graph of function g has a sharp corner. The graph of function k is not continuous. Q & A Do all polnomial functions have as their domain all real numbers? Yes. An real number is a valid input for a polnomial function. Using Factoring to Find Zeros of Polnomial Functions Recall that if f is a polnomial function, the values of for hich f () = 0 are called zeros of f. If the equation of the polnomial function can be factored, e can set each factor equal to zero and solve for the zeros. We can use this method to find -intercepts because at the -intercepts e find the input values hen the output value is zero. For general polnomials, this can be a challenging prospect. While quadratics can be solved using the relativel simple quadratic formula, the corresponding formulas for cubic and fourth-degree polnomials are not simple enough to remember, and formulas do not eist for general higher-degree polnomials. Consequentl, e ill limit ourselves to three cases:. The polnomial can be factored using knon methods: greatest common factor and trinomial factoring.. The polnomial is given in factored form.. Technolog is used to determine the intercepts. Ho To Given a polnomial function f, find the -intercepts b factoring.. Set f () = 0.. If the polnomial function is not given in factored form: a. Factor out an common monomial factors. b. Factor an factorable binomials or trinomials.. Set each factor equal to zero and solve to find the -intercepts. Donload for free at http://cn.org/contents/9b08c9-07f-0-9f8-dad9970d@.9

SECTION. GRAPHS OF POLYNOMIAL FUNCTIONS 77 Eample Finding the -Intercepts of a Polnomial Function b Factoring Find the -intercepts of f () = +. Solution We can attempt to factor this polnomial to find solutions for f () = 0. + = 0 ( + ) = 0 ( )( ) = 0 Factor out the greatest common factor. Factor the trinomial. ( ) = 0 ( ) = 0 Set each factor equal to zero. = 0 or = or = = 0 = ± = ± This gives us five -intercepts: (0, 0), (, 0), (, 0), (, 0 ), and (, 0 ). See Figure. We can see that this is an even function because it is smmetric about the -ais. f () = + (, 0) (, 0) (0, 0) (, 0) (, 0) Eample Figure Finding the -Intercepts of a Polnomial Function b Factoring Find the -intercepts of f () = +. Solution Find solutions for f () = 0 b factoring. + = 0 Factor b grouping. ( ) ( ) = 0 Factor out the common factor. ( )( ) = 0 Factor the difference of squares. ( + )( )( ) = 0 Set each factor equal to zero. + = 0 or = 0 or = 0 = = = There are three -intercepts: (, 0), (, 0), and (, 0). See Figure. 8 f () = + 8 Figure Donload for free at http://cn.org/contents/9b08c9-07f-0-9f8-dad9970d@.9

78 CHAPTER POLYNOMIAL AND RATIONAL FUNCTIONS Eample Finding the - and -Intercepts of a Polnomial in Factored Form Find the - and -intercepts of g() = ( ) ( + ). Solution The -intercept can be found b evaluating g(0). So the -intercept is (0, ). The -intercepts can be found b solving g() = 0. So the -intercepts are (, 0) and (, 0 ). g(0) = (0 ) ((0) + ) = ( ) ( + ) = 0 ( ) = 0 ( + ) = 0 = 0 or = = Analsis We can alas check that our ansers are reasonable b using a graphing calculator to graph the polnomial as shon in Figure. g() = ( ) ( + ) (0, ) 9 (., 0) (, 0) Figure Eample Finding the -Intercepts of a Polnomial Function Using a Graph Find the -intercepts of h() = + +. Solution This polnomial is not in factored form, has no common factors, and does not appear to be factorable using techniques previousl discussed. Fortunatel, e can use technolog to find the intercepts. Keep in mind that some values make graphing difficult b hand. In these cases, e can take advantage of graphing utilities. Looking at the graph of this function, as shon in Figure, it appears that there are -intercepts at =,, and. h() = + + 8 Figure Donload for free at http://cn.org/contents/9b08c9-07f-0-9f8-dad9970d@.9

SECTION. GRAPHS OF POLYNOMIAL FUNCTIONS 79 We can check hether these are correct b substituting these values for and verifing that Since h() = + +, e have: h( ) = h( ) = h() = 0. h( ) = ( ) + ( ) + ( ) = 7 + = 0 h( ) = ( ) + ( ) + ( ) = 8 + = 0 h() = () + () + () = + + = 0 Each -intercept corresponds to a zero of the polnomial function and each zero ields a factor, so e can no rite the polnomial in factored form. h() = + + = ( + )( + )( ) Tr It # Find the - and -intercepts of the function f () = 9 + 0. Identifing Zeros and Their Multiplicities Graphs behave differentl at various -intercepts. Sometimes, the graph ill cross over the horizontal ais at an intercept. Other times, the graph ill touch the horizontal ais and "bounce" off. Suppose, for eample, e graph the function shon. f () = ( + )( ) ( + ). Notice in Figure 7 that the behavior of the function at each of the -intercepts is different. 0 0 0 0 0 0 0 0 f () = ( + )( ) ( + ) Figure 7 Identifing the behavior of the graph at an -intercept b eamining the multiplicit of the zero. The -intercept = is the solution of equation ( + ) = 0. The graph passes directl through the -intercept at =. The factor is linear (has a degree of ), so the behavior near the intercept is like that of a line it passes directl through the intercept. We call this a single zero because the zero corresponds to a single factor of the function. The -intercept = is the repeated solution of equation ( ) = 0. The graph touches the ais at the intercept and changes direction. The factor is quadratic (degree ), so the behavior near the intercept is like that of a quadratic it bounces off of the horizontal ais at the intercept. ( ) = ( )( ) The factor is repeated, that is, the factor ( ) appears tice. The number of times a given factor appears in the factored form of the equation of a polnomial is called the multiplicit. The zero associated ith this factor, =, has multiplicit because the factor ( ) occurs tice. Donload for free at http://cn.org/contents/9b08c9-07f-0-9f8-dad9970d@.9

80 CHAPTER POLYNOMIAL AND RATIONAL FUNCTIONS The -intercept = is the repeated solution of factor ( + ) = 0. The graph passes through the ais at the intercept, but flattens out a bit first. This factor is cubic (degree ), so the behavior near the intercept is like that of a cubic ith the same S-shape near the intercept as the toolkit function f () =. We call this a triple zero, or a zero ith multiplicit. For zeros ith even multiplicities, the graphs touch or are tangent to the -ais. For zeros ith odd multiplicities, the graphs cross or intersect the -ais. See Figure 8 for eamples of graphs of polnomial functions ith multiplicit,, and. p = p = p = Single zero Zero ith multiplicit Figure 8 Zero ith multiplicit For higher even poers, such as,, and 8, the graph ill still touch and bounce off of the horizontal ais but, for each increasing even poer, the graph ill appear flatter as it approaches and leaves the -ais. For higher odd poers, such as, 7, and 9, the graph ill still cross through the horizontal ais, but for each increasing odd poer, the graph ill appear flatter as it approaches and leaves the -ais. graphical behavior of polnomials at -intercepts If a polnomial contains a factor of the form ( h) p, the behavior near the -intercept h is determined b the poer p. We sa that = h is a zero of multiplicit p. The graph of a polnomial function ill touch the -ais at zeros ith even multiplicities. The graph ill cross the -ais at zeros ith odd multiplicities. The sum of the multiplicities is the degree of the polnomial function. Ho To Given a graph of a polnomial function of degree n, identif the zeros and their multiplicities.. If the graph crosses the -ais and appears almost linear at the intercept, it is a single zero.. If the graph touches the -ais and bounces off of the ais, it is a zero ith even multiplicit.. If the graph crosses the -ais at a zero, it is a zero ith odd multiplicit.. The sum of the multiplicities is n. Eample Identifing Zeros and Their Multiplicities Use the graph of the function of degree in Figure 9 to identif the zeros of the function and their possible multiplicities. Donload for free at http://cn.org/contents/9b08c9-07f-0-9f8-dad9970d@.9

SECTION. GRAPHS OF POLYNOMIAL FUNCTIONS 8 0 0 80 80 0 0 Figure 9 Solution The polnomial function is of degree. The sum of the multiplicities must be. Starting from the left, the first zero occurs at =. The graph touches the -ais, so the multiplicit of the zero must be even. The zero of most likel has multiplicit. The net zero occurs at =. The graph looks almost linear at this point. This is a single zero of multiplicit. The last zero occurs at =. The graph crosses the -ais, so the multiplicit of the zero must be odd. We kno that the multiplicit is likel and that the sum of the multiplicities is likel. Tr It # Use the graph of the function of degree in Figure 0 to identif the zeros of the function and their multiplicities. 0 0 0 0 0 0 Figure 0 Determining End Behavior As e have alread learned, the behavior of a graph of a polnomial function of the form f () = a n n + a n n +... + a + a 0 ill either ultimatel rise or fall as increases ithout bound and ill either rise or fall as decreases ithout bound. This is because for ver large inputs, sa 00 or,000, the leading term dominates the size of the output. The same is true for ver small inputs, sa 00 or,000. Recall that e call this behavior the end behavior of a function. As e pointed out hen discussing quadratic equations, hen the leading term of a polnomial function, a n n, is an even poer function, as increases or decreases ithout bound, f () increases ithout bound. When the leading term is an odd poer function, as decreases ithout bound, f () also decreases ithout bound; as increases ithout bound, f () also increases ithout bound. If the leading term is negative, it ill change the direction of the end behavior. Figure summarizes all four cases. Donload for free at http://cn.org/contents/9b08c9-07f-0-9f8-dad9970d@.9

8 CHAPTER POLYNOMIAL AND RATIONAL FUNCTIONS Even Degree Positive Leading Coefficient, a n > 0 Odd Degree Positive Leading Coefficient, a n > 0 End Behavior:, f (), f () Negative Leading Coefficient, a n < 0 End Behavior:, f (), f () Negative Leading Coefficient, a n < 0 End Behavior:, f (), f () End Behavior:, f (), f () Figure Understanding the Relationship Beteen Degree and Turning Points In addition to the end behavior, recall that e can analze a polnomial function s local behavior. It ma have a turning point here the graph changes from increasing to decreasing (rising to falling) or decreasing to increasing (falling to rising). Look at the graph of the polnomial function f () = + in Figure. The graph has three turning points. Increasing Decreasing Increasing Decreasing Turning points Figure This function f is a th degree polnomial function and has turning points. The maimum number of turning points of a polnomial function is alas one less than the degree of the function. Donload for free at http://cn.org/contents/9b08c9-07f-0-9f8-dad9970d@.9

SECTION. GRAPHS OF POLYNOMIAL FUNCTIONS 8 interpreting turning points A turning point is a point of the graph here the graph changes from increasing to decreasing (rising to falling) or decreasing to increasing (falling to rising). A polnomial of degree n ill have at most n turning points. Eample 7 Finding the Maimum Number of Turning Points Using the Degree of a Polnomial Function Find the maimum number of turning points of each polnomial function. a. f () = + + b. f () = ( ) ( + ) Solution a. f () = + + First, rerite the polnomial function in descending order: f () = + Identif the degree of the polnomial function. This polnomial function is of degree. The maimum number of turning points is =. b. f () = ( ) ( + ) First, identif the leading term of the polnomial function if the function ere epanded. f () = ( ) ( + ) a n = ( )( ) Then, identif the degree of the polnomial function. This polnomial function is of degree. The maimum number of turning points is =. Graphing Polnomial Functions We can use hat e have learned about multiplicities, end behavior, and turning points to sketch graphs of polnomial functions. Let us put this all together and look at the steps required to graph polnomial functions. Ho To Given a polnomial function, sketch the graph.. Find the intercepts.. Check for smmetr. If the function is an even function, its graph is smmetrical about the -ais, that is, f ( ) = f (). If a function is an odd function, its graph is smmetrical about the origin, that is, f ( ) = f ().. Use the multiplicities of the zeros to determine the behavior of the polnomial at the -intercepts.. Determine the end behavior b eamining the leading term.. Use the end behavior and the behavior at the intercepts to sketch a graph.. Ensure that the number of turning points does not eceed one less than the degree of the polnomial. 7. Optionall, use technolog to check the graph. Eample 8 Sketching the Graph of a Polnomial Function Sketch a graph of f () = ( + ) ( ). Solution This graph has to -intercepts. At =, the factor is squared, indicating a multiplicit of. The graph ill bounce at this -intercept. At =, the function has a multiplicit of one, indicating the graph ill cross through the ais at this intercept. The -intercept is found b evaluating f (0). The -intercept is (0, 90). f (0) = (0 + ) (0 ) = ċ 9 ċ ( ) = 90 Donload for free at http://cn.org/contents/9b08c9-07f-0-9f8-dad9970d@.9

8 CHAPTER POLYNOMIAL AND RATIONAL FUNCTIONS Additionall, e can see the leading term, if this polnomial ere multiplied out, ould be, so the end behavior is that of a verticall reflected cubic, ith the outputs decreasing as the inputs approach infinit, and the outputs increasing as the inputs approach negative infinit. See Figure. To sketch this, e consider that: Figure As the function f (), so e kno the graph starts in the second quadrant and is decreasing toard the -ais. Since f ( ) = ( + ) ( ) is not equal to f (), the graph does not displa smmetr. At (, 0), the graph bounces off of the -ais, so the function must start increasing. At (0, 90), the graph crosses the -ais at the -intercept. See Figure. (0, 90) (, 0) Figure Somehere after this point, the graph must turn back don or start decreasing toard the horizontal ais because the graph passes through the net intercept at (, 0). See Figure. (0, 90) (, 0) (, 0) Figure As the function f (), so e kno the graph continues to decrease, and e can stop draing the graph in the fourth quadrant. Donload for free at http://cn.org/contents/9b08c9-07f-0-9f8-dad9970d@.9

SECTION. GRAPHS OF POLYNOMIAL FUNCTIONS 8 Using technolog, e can create the graph for the polnomial function, shon in Figure, and verif that the resulting graph looks like our sketch in Figure. 80 f () = ( + ) ( ) 0 0 0 0 80 Figure The complete graph of the polnomial function f ( ) = ( + ) ( ) Tr It # Sketch a graph of f () = ( ) ( + ). Using the Intermediate Value Theorem In some situations, e ma kno to points on a graph but not the zeros. If those to points are on opposite sides of the -ais, e can confirm that there is a zero beteen them. Consider a polnomial function f hose graph is smooth and continuous. The Intermediate Value Theorem states that for to numbers a and b in the domain of f, if a < b and f (a) f (b), then the function f takes on ever value beteen f (a) and f (b). (While the theorem is intuitive, the proof is actuall quite complicated and require higher mathematics.) We can appl this theorem to a special case that is useful in graphing polnomial functions. If a point on the graph of a continuous function f at = a lies above the -ais and another point at = b lies belo the -ais, there must eist a third point beteen = a and = b here the graph crosses the -ais. Call this point (c, f (c)). This means that e are assured there is a solution c here f (c) = 0. In other ords, the Intermediate Value Theorem tells us that hen a polnomial function changes from a negative value to a positive value, the function must cross the -ais. Figure 7 shos that there is a zero beteen a and b. f (b) is positive f (c) = 0 f(a) is negative Figure 7 Using the Intermediate Value Theorem to sho there eists a zero Intermediate Value Theorem Let f be a polnomial function. The Intermediate Value Theorem states that if f (a) and f (b) have opposite signs, then there eists at least one value c beteen a and b for hich f (c) = 0. Donload for free at http://cn.org/contents/9b08c9-07f-0-9f8-dad9970d@.9

8 CHAPTER POLYNOMIAL AND RATIONAL FUNCTIONS Eample 9 Using the Intermediate Value Theorem Sho that the function f () = + + has at least to real zeros beteen = and =. Solution As a start, evaluate f () at the integer values =,,, and. See Table. f () 0 Table We see that one zero occurs at =. Also, since f () is negative and f () is positive, b the Intermediate Value Theorem, there must be at least one real zero beteen and. We have shon that there are at least to real zeros beteen = and =. Analsis We can also see on the graph of the function in Figure 8 that there are to real zeros beteen = and =. 0 8 8 0 f () = is positive f () = is negative f () = is positive Figure 8 Tr It # Sho that the function f () = 7 9 has at least one real zero beteen = and =. Writing Formulas for Polnomial Functions No that e kno ho to find zeros of polnomial functions, e can use them to rite formulas based on graphs. Because a polnomial function ritten in factored form ill have an -intercept here each factor is equal to zero, e can form a function that ill pass through a set of -intercepts b introducing a corresponding set of factors. factored form of polnomials If a polnomial of loest degree p has horizontal intercepts at =,,, n, then the polnomial can be ritten in the factored form: f () = a( ) p ( ) p ( n) p n here the poers p i on each factor can be determined b the behavior of the graph at the corresponding intercept, and the stretch factor a can be determined given a value of the function other than the -intercept. Ho To Given a graph of a polnomial function, rite a formula for the function.. Identif the -intercepts of the graph to find the factors of the polnomial.. Eamine the behavior of the graph at the -intercepts to determine the multiplicit of each factor.. Find the polnomial of least degree containing all the factors found in the previous step.. Use an other point on the graph (the -intercept ma be easiest) to determine the stretch factor. Donload for free at http://cn.org/contents/9b08c9-07f-0-9f8-dad9970d@.9

SECTION. GRAPHS OF POLYNOMIAL FUNCTIONS 87 Eample 0 Writing a Formula for a Polnomial Function from the Graph Write a formula for the polnomial function shon in Figure 9. Figure 9 Solution This graph has three -intercepts: =,, and. The -intercept is located at (0, ). At = and =, the graph passes through the ais linearl, suggesting the corresponding factors of the polnomial ill be linear. At =, the graph bounces at the intercept, suggesting the corresponding factor of the polnomial ill be second degree (quadratic). Together, this gives us f () = a( + )( ) ( ) To determine the stretch factor, e utilize another point on the graph. We ill use the -intercept (0, ), to solve for a. f (0) = a(0 + )(0 ) (0 ) = a(0 + )(0 ) (0 ) = 0a a = 0 The graphed polnomial appears to represent the function f () = 0 ( + )( ) ( ). Tr It # Given the graph shon in Figure 0, rite a formula for the function shon. 8 8 Figure 0 Using Local and Global Etrema With quadratics, e ere able to algebraicall find the maimum or minimum value of the function b finding the verte. For general polnomials, finding these turning points is not possible ithout more advanced techniques from calculus. Even then, finding here etrema occur can still be algebraicall challenging. For no, e ill estimate the locations of turning points using technolog to generate a graph. Each turning point represents a local minimum or maimum. Sometimes, a turning point is the highest or loest point on the entire graph. In these cases, e sa that the turning point is a global maimum or a global minimum. These are also referred to as the absolute maimum and absolute minimum values of the function. Donload for free at http://cn.org/contents/9b08c9-07f-0-9f8-dad9970d@.9

88 CHAPTER POLYNOMIAL AND RATIONAL FUNCTIONS local and global etrema A local maimum or local minimum at = a (sometimes called the relative maimum or minimum, respectivel) is the output at the highest or loest point on the graph in an open interval around = a. If a function has a local maimum at a, then f (a) f () for all in an open interval around = a. If a function has a local minimum at a, then f (a) f () for all in an open interval around = a. A global maimum or global minimum is the output at the highest or loest point of the function. If a function has a global maimum at a, then f (a) f () for all. If a function has a global minimum at a, then f (a) f () for all. We can see the difference beteen local and global etrema in Figure. Global maimum Local maimum Figure Local minimum Q & A Do all polnomial functions have a global minimum or maimum? No. Onl polnomial functions of even degree have a global minimum or maimum. For eample, f () = has neither a global maimum nor a global minimum. Eample Using Local Etrema to Solve Applications An open-top bo is to be constructed b cutting out squares from each corner of a cm b 0 cm sheet of plastic then folding up the sides. Find the size of squares that should be cut out to maimize the volume enclosed b the bo. Solution We ill start this problem b draing a picture like that in Figure, labeling the idth of the cut-out squares ith a variable,. Figure Notice that after a square is cut out from each end, it leaves a ( ) cm b (0 ) cm rectangle for the base of the bo, and the bo ill be cm tall. This gives the volume V() = (0 )( ) = 80 8 + Donload for free at http://cn.org/contents/9b08c9-07f-0-9f8-dad9970d@.9

SECTION. GRAPHS OF POLYNOMIAL FUNCTIONS 89 Notice, since the factors are, 0 and, the three zeros are 0, 7, and 0, respectivel. Because a height of 0 cm is not reasonable, e consider the onl the zeros 0 and 7. The shortest side is and e are cutting off to squares, so values ma take on are greater than zero or less than 7. This means e ill restrict the domain of this function to 0 < < 7. Using technolog to sketch the graph of V() on this reasonable domain, e get a graph like that in Figure. We can use this graph to estimate the maimum value for the volume, restricted to values for that are reasonable for this problem values from 0 to 7. V() V() = 80 8 + 00 00 00 00 8 0 00 00 Figure From this graph, e turn our focus to onl the portion on the reasonable domain, [0, 7]. We can estimate the maimum value to be around 0 cubic cm, hich occurs hen the squares are about.7 cm on each side. To improve this estimate, e could use advanced features of our technolog, if available, or simpl change our indo to zoom in on our graph to produce Figure. 0 9 8 7 V() 0...8 Figure From this zoomed-in vie, e can refine our estimate for the maimum volume to about 9 cubic cm, hen the squares measure approimatel.7 cm on each side. Tr It # Use technolog to find the maimum and minimum values on the interval [, ] of the function f () = 0.( ) ( + ) ( ). Access the folloing online resource for additional instruction and practice ith graphing polnomial functions. Intermediate Value Theorem (http://openstacollege.org/l/ivt) Donload for free at http://cn.org/contents/9b08c9-07f-0-9f8-dad9970d@.9

90 CHAPTER POLYNOMIAL AND RATIONAL FUNCTIONS. SECTION EXERCISES VERBAL. What is the difference beteen an -intercept and a zero of a polnomial function f?. Eplain ho the Intermediate Value Theorem can assist us in finding a zero of a function.. If the graph of a polnomial just touches the -ais and then changes direction, hat can e conclude about the factored form of the polnomial?. If a polnomial function of degree n has n distinct zeros, hat do ou kno about the graph of the function?. Eplain ho the factored form of the polnomial helps us in graphing it. ALGEBRAIC For the folloing eercises, find the - or t-intercepts of the polnomial functions.. C(t) = (t )(t + )(t ) 7. C(t) = (t + )(t )(t + ) 8. C(t) = t(t ) (t + ) 9. C(t) = t(t )(t + ) 0. C(t) = t 8t + t. C(t) = t + t 0t. f () =. f () = + 0. f () = + 7. f () = +. f () = + 9 8 7. f () = 8 + 8. f () = 7 8 9. f () = + 8 0. f () = +. f () =. f () =. f () = + For the folloing eercises, use the Intermediate Value Theorem to confirm that the given polnomial has at least one zero ithin the given interval.. f () = 9, beteen = and =.. f () = 9, beteen = and =.. f () =, beteen = and =. 7. f () = +, beteen = and =. 8. f () =, beteen = and =. 9. f () = 00 +, beteen = 0.0 and = 0. For the folloing eercises, find the zeros and give the multiplicit of each. 0. f () = ( + ) ( ). f () = ( + ) ( ). f () = ( ) ( + ). f () = ( + + ). f () = ( + ) (9 + ). f () = ( + ) ( 0 + ). f () = ( + 9)( + 8 + ) 7. f () = 8. f () = + + 9. f () = + 9 0. f () = ( + ). f () = (9 + ) GRAPHICAL For the folloing eercises, graph the polnomial functions. Note - and -intercepts, multiplicit, and end behavior.. f () = ( + ) ( ). g() = ( + )( ). h() = ( ) ( + ). k() = ( ) ( ). m() = ( )( + ) 7. n() = ( + )( ) Donload for free at http://cn.org/contents/9b08c9-07f-0-9f8-dad9970d@.9

SECTION. SECTION EXERCISES 9 For the folloing eercises, use the graphs to rite the formula for a polnomial function of least degree. 8. f() 9. f() 0. f(). f(). f() For the folloing eercises, use the graph to identif zeros and multiplicit..... For the folloing eercises, use the given information about the polnomial graph to rite the equation. 7. Degree. Zeros at =, =, and =. -intercept at (0, ). 9. Degree. Roots of multiplicit at = and =, and a root of multiplicit at =. -intercept at (0, 9) 8. Degree. Zeros at =, =, and =. -intercept at (0, ) 0. Degree. Root of multiplicit at =, and roots of multiplicit at = and =. -intercept at (0, ). Donload for free at http://cn.org/contents/9b08c9-07f-0-9f8-dad9970d@.9

9 CHAPTER POLYNOMIAL AND RATIONAL FUNCTIONS. Degree. Double zero at =, and triple zero at =. Passes through the point (, ).. Degree. Zeros at =, = and =. -intercept at (0, ).. Degree. Roots of multiplicit at = _ and roots of multiplicit at = and =. -intercept at (0,8).. Degree. Zeros at =, =, and =. -intercept at (0, ).. Degree. Roots of multiplicit at = and = and a root of multiplicit at =. -intercept at (0, ).. Double zero at = and triple zero at = 0. Passes through the point (, ). TECHNOLOGY For the folloing eercises, use a calculator to approimate local minima and maima or the global minimum and maimum. 7. f () = 8. f () = 9. f () = + 70. f () = + 7. f () = + EXTENSIONS For the folloing eercises, use the graphs to rite a polnomial function of least degree. 7. 7. f (), 0 (0, 8) ( 00, 0) 8 0 0 f(), 0 (00, 0) 00 00 00 00 00 00 0 (0, 90,000) 7. 0 7 0 7 0 7 0 7 0 7 0 7 f () 00 00 00 0 00 0 7 0 7 0 7 0 7 0 7 0 7 7 0 7 (0, 0,000,000) 00 00 00 00 00 700 0 0 0 REAL-WORLD APPLICATIONS For the folloing eercises, rite the polnomial function that models the given situation. 7. A rectangle has a length of 0 units and a idth of 8 units. Squares of b units are cut out of each corner, and then the sides are folded up to create an open bo. Epress the volume of the bo as a polnomial function in terms of. 77. A square has sides of units. Squares + b + units are cut out of each corner, and then the sides are folded up to create an open bo. Epress the volume of the bo as a function in terms of. 79. A right circular cone has a radius of + and a height units less. Epress the volume of the cone as a polnomial function. The volume of a cone is V = _ πr h for radius r and height h. Donload for free at http://cn.org/contents/9b08c9-07f-0-9f8-dad9970d@.9 7. Consider the same rectangle of the preceding problem. Squares of b units are cut out of each corner. Epress the volume of the bo as a polnomial in terms of. 78. A clinder has a radius of + units and a height of units greater. Epress the volume of the clinder as a polnomial function.