Zeros of Polynomial Functions. The Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra. zero in the complex number system.

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1 _.qd /7/ 9:6 AM Page 69 Section. Zeros of Polnomial Functions 69. Zeros of Polnomial Functions What ou should learn Use the Fundamental Theorem of Algebra to determine the number of zeros of polnomial functions. Find rational zeros of polnomial functions. Find conjugate pairs of comple zeros. Find zeros of polnomials b factoring. Use Descartes s Rule of Signs and the Upper and Lower Bound Rules to find zeros of polnomials. Wh ou should learn it Finding zeros of polnomial functions is an important part of solving real-life problems. For instance, in Eercise on page 8, the zeros of a polnomial function can help ou analze the attendance at women s college basketball games. The Fundamental Theorem of Algebra You know that an nth-degree polnomial can have at most n real zeros. In the comple number sstem, this statement can be improved. That is, in the comple number sstem, ever nth-degree polnomial function has precisel n zeros. This important result is derived from the Fundamental Theorem of Algebra, first proved b the German mathematician Carl Friedrich Gauss (777 8). The Fundamental Theorem of Algebra If f is a polnomial of degree n, where n >, then f has at least one zero in the comple number sstem. Using the Fundamental Theorem of Algebra and the equivalence of zeros and factors, ou obtain the Linear Factorization Theorem. Linear Factorization Theorem If f is a polnomial of degree n, where n >, then f has precisel n linear factors f a n c c... c n where c, c,..., c n are comple numbers. For a proof of the Linear Factorization Theorem, see Proofs in Mathematics on page 4. Note that the Fundamental Theorem of Algebra and the Linear Factorization Theorem tell ou onl that the zeros or factors of a polnomial eist, not how to find them. Such theorems are called eistence theorems. Eample Zeros of Polnomial Functions Recall that in order to find the zeros of a function f, set f equal to and solve the resulting equation for. For instance, the function in Eample (a) has a zero at because. a. The first-degree polnomial f has eactl one zero:. b. Counting multiplicit, the second-degree polnomial function f 6 9 has eactl two zeros: and. (This is called a repeated zero.) c. The third-degree polnomial function f 4 4 i i has eactl three zeros:, i, and i. d. The fourth-degree polnomial function f 4 i i has eactl four zeros:,, i, and i. Now tr Eercise.

2 _.qd /7/ 9:6 AM Page 7 7 Chapter Polnomial and Rational Functions Finding zeros of polnomial functions is a ver important concept in algebra. This is a good place to discuss the fact that polnomials do not necessaril have rational zeros but ma have zeros that are irrational or comple. The Rational Zero Test The Rational Zero Test relates the possible rational zeros of a polnomial (having integer coefficients) to the leading coefficient and to the constant term of the polnomial. The Rational Zero Test If the polnomial f a n n a n n... a a a has integer coefficients, ever rational zero of f has the form Fogg Art Museum Historical Note Although the were not contemporaries,jean Le Rond d Alembert (77 78) worked independentl of Carl Gauss in tring to prove the Fundamental Theorem of Algebra. His efforts were such that, in France, the Fundamental Theorem of Algebra is frequentl known as the Theorem of d Alembert. Rational zero p q where p and q have no common factors other than, and p a factor of the constant term a q a factor of the leading coefficient a n. To use the Rational Zero Test, ou should first list all rational numbers whose numerators are factors of the constant term and whose denominators are factors of the leading coefficient. Possible rational zeros factors of constant term factors of leading coefficient Having formed this list of possible rational zeros, use a trial-and-error method to determine which, if an, are actual zeros of the polnomial. Note that when the leading coefficient is, the possible rational zeros are simpl the factors of the constant term. Eample Rational Zero Test with Leading Coefficient of f() = + + FIGURE. Find the rational zeros of f. Because the leading coefficient is, the possible rational zeros are ±, the factors of the constant term. B testing these possible zeros, ou can see that neither works. f f So, ou can conclude that the given polnomial has no rational zeros. Note from the graph of f in Figure. that f does have one real zero between and. However, b the Rational Zero Test, ou know that this real zero is not a rational number. Now tr Eercise 7.

3 _.qd /7/ 9:6 AM Page 7 Section. Zeros of Polnomial Functions 7 Eample Rational Zero Test with Leading Coefficient of When the list of possible rational zeros is small, as in Eample, it ma be quicker to test the zeros b evaluating the function. When the list of possible rational zeros is large, as in Eample, it ma be quicker to use a different approach to test the zeros, such as using snthetic division or sketching a graph. Find the rational zeros of f 4 6. Because the leading coefficient is, the possible rational zeros are the factors of the constant term. Possible rational zeros: ±, ±, ±, ±6 B appling snthetic division successivel, ou can determine that and are the onl two rational zeros. So, f factors as f remainder, so is a zero. remainder, so is a zero. Because the factor produces no real zeros, ou can conclude that and are the onl real zeros of f, which is verified in Figure.. Additional Eample List the possible rational zeros of f 8 4. The leading coefficient is, so the possible rational zeros are ±, ±, ±, ±7, ±, ±, ±, ±, ±7, ±, ±7, and ±. To decide which possible rational zeros should be tested using snthetic division, graph the function. From the graph, ou can see that the zero is positive and less than, so the onl values of that should be tested are,,, and FIGURE. Now tr Eercise. 8 6 (, ) (, ) f () = + 6 If the leading coefficient of a polnomial is not, the list of possible rational zeros can increase dramaticall. In such cases, the search can be shortened in several was: () a programmable calculator can be used to speed up the calculations; () a graph, drawn either b hand or with a graphing utilit, can give a good estimate of the locations of the zeros; () the Intermediate Value Theorem along with a table generated b a graphing utilit can give approimations of zeros; and (4) snthetic division can be used to test the possible rational zeros. Finding the first zero is often the most difficult part. After that, the search is simplified b working with the lower-degree polnomial obtained in snthetic division, as shown in Eample. 8 4

4 _.qd /7/ 9:6 AM Page 7 7 Chapter Polnomial and Rational Functions Eample 4 Using the Rational Zero Test Remember that when ou tr to find the rational zeros of a polnomial function with man possible rational zeros, as in Eample 4, ou must use trial and error. There is no quick algebraic method to determine which of the possibilities is an actual zero; however, sketching a graph ma be helpful. Find the rational zeros of f 8. The leading coefficient is and the constant term is. Factors of ±, ± Possible rational zeros: Factors of ±, ± ±, ±, ±, ± B snthetic division, ou can determine that is a rational zero. So, f factors as 8 f and ou can conclude that the rational zeros of f are,, and. Now tr Eercise 7. Recall from Section. that if a is a zero of the polnomial function then a is a solution of the polnomial equation f. f, f () = FIGURE. Eample Solving a Polnomial Equation Find all the real solutions of 6. The leading coefficient is and the constant term is. Factors of ±, ±, ±, ±4, ±6, ± Possible rational solutions: Factors of ±, ±, ±, ± With so man possibilities (, in fact), it is worth our time to stop and sketch a graph. From Figure., it looks like three reasonable solutions would be 6,, and. Testing these b snthetic division shows that is the onl rational solution. So, ou have Using the Quadratic Formula for the second factor, ou find that the two additional solutions are irrational numbers. and Now tr Eercise.

5 _.qd /7/ 9:6 AM Page 7 Conjugate Pairs Section. Zeros of Polnomial Functions 7 In Eample (c) and (d), note that the pairs of comple zeros are conjugates. That is, the are of the form a bi and a bi. Comple Zeros Occur in Conjugate Pairs Let f be a polnomial function that has real coefficients. If a bi, where b, is a zero of the function, the conjugate a bi is also a zero of the function. Be sure ou see that this result is true onl if the polnomial function has real coefficients. For instance, the result applies to the function given b f but not to the function given b g i. Eample 6 Finding a Polnomial with Given Zeros Find a fourth-degree polnomial function with real coefficients that has,, and i as zeros. Because i is a zero and the polnomial is stated to have real coefficients, ou know that the conjugate i must also be a zero. So, from the Linear Factorization Theorem, f can be written as f a i i. For simplicit, let a to obtain f Now tr Eercise 7. Factoring a Polnomial The Linear Factorization Theorem shows that ou can write an nth-degree polnomial as the product of n linear factors. f a n c c c... c n However, this result includes the possibilit that some of the values of are comple. The following theorem sas that even if ou do not want to get involved with comple factors, ou can still write f as the product of linear and/or quadratic factors. For a proof of this theorem, see Proofs in Mathematics on page 4. Factors of a Polnomial Ever polnomial of degree n > with real coefficients can be written as the product of linear and quadratic factors with real coefficients, where the quadratic factors have no real zeros. c i

6 _.qd /7/ 9:6 AM Page Chapter Polnomial and Rational Functions You ma want to remind students that a graphing calculator is helpful in determining real zeros, which in turn are useful in finding the comple zeros. A quadratic factor with no real zeros is said to be prime or irreducible over the reals. Be sure ou see that this is not the same as being irreducible over the rationals. For eample, the quadratic i i is irreducible over the reals (and therefore over the rationals). On the other hand, the quadratic is irreducible over the rationals but reducible over the reals. Eample 7 Finding the Zeros of a Polnomial Function Find all the zeros of f given that i is a zero of f. Algebraic Because comple zeros occur in conjugate pairs, ou know that i is also a zero of f. This means that both i and i are factors of f. Multipling these two factors produces i i i i Using long division, ou can divide into f to obtain the following. ) So, ou have 4 4 f i. 6 and ou can conclude that the zeros of i,, and. Now tr Eercise 47. f are i, Graphical Because comple zeros alwas occur in conjugate pairs, ou know that i is also a zero of f. Because the polnomial is a fourth-degree polnomial, ou know that there are at most two other zeros of the function. Use a graphing utilit to graph as shown in Figure.. = FIGURE. You can see that and appear to be zeros of the graph of the function. Use the zero or root feature or the zoom and trace features of the graphing utilit to confirm that and are zeros of the graph. So, ou can conclude that the zeros of f are i, i,, and. In Eample 7, if ou were not told that i is a zero of f, ou could still find all zeros of the function b using snthetic division to find the real zeros and. Then ou could factor the polnomial as. Finall, b using the Quadratic Formula, ou could determine that the zeros are,, i, and i. 8 8

7 _.qd /7/ 9:6 AM Page 7 Section. Zeros of Polnomial Functions 7 Eample 8 shows how to find all the zeros of a polnomial function, including comple zeros. In Eample 8, the fifth-degree polnomial function has three real zeros. In such cases, ou can use the zoom and trace features or the zero or root feature of a graphing utilit to approimate the real zeros. You can then use these real zeros to determine the comple zeros algebraicall. f() = (, ) (, ) 4 4 FIGURE.4 Eample 8 Finding the Zeros of a Polnomial Function Write f 8 as the product of linear factors, and list all of its zeros. The possible rational zeros are ±, ±, ±4, and ±8. Snthetic division produces the following. So, ou have f 8 You can factor 4 4 as 4, and b factoring 4 as ou obtain i i f i i is a zero. is a zero. which gives the following five zeros of f.,,, i, and i From the graph of f shown in Figure.4, ou can see that the real zeros are the onl ones that appear as -intercepts. Note that is a repeated zero. Now tr Eercise 6. You can use the table feature of a graphing utilit to help ou determine which of the possible rational zeros are zeros of the polnomial in Eample 8. The table should be set to ask mode. Then enter each of the possible rational zeros in the table. When ou do this, ou will see that there are two rational zeros, and, as shown at the right. Technolog

8 _.qd /7/ 9:6 AM Page Chapter Polnomial and Rational Functions Other Tests for Zeros of Polnomials You know that an nth-degree polnomial function can have at most n real zeros. Of course, man nth-degree polnomials do not have that man real zeros. For instance, f has no real zeros, and f has onl one real zero. The following theorem, called Descartes s Rule of Signs, sheds more light on the number of real zeros of a polnomial. Descartes s Rule of Signs Let f () a n n a n n... a a a be a polnomial with real coefficients and a.. The number of positive real zeros of f is either equal to the number of variations in sign of f or less than that number b an even integer.. The number of negative real zeros of f is either equal to the number of variations in sign of f or less than that number b an even integer. A variation in sign means that two consecutive coefficients have opposite signs. When using Descartes s Rule of Signs, a zero of multiplicit k should be counted as k zeros. For instance, the polnomial has two variations in sign, and so has either two positive or no positive real zeros. Because ou can see that the two positive real zeros are of multiplicit. Eample 9 Using Descartes s Rule of Signs Describe the possible real zeros of f 6 4. The original polnomial has three variations in sign. to to f() = f 6 4 FIGURE. to The polnomial f has no variations in sign. So, from Descartes s Rule of Signs, the polnomial f 6 4 has either three positive real zeros or one positive real zero, and has no negative real zeros. From the graph in Figure., ou can see that the function has onl one real zero (it is a positive number, near ). Now tr Eercise 79.

9 _.qd /7/ 9:6 AM Page 77 Section. Zeros of Polnomial Functions 77 Another test for zeros of a polnomial function is related to the sign pattern in the last row of the snthetic division arra. This test can give ou an upper or lower bound of the real zeros of f. A real number b is an upper bound for the real zeros of f if no zeros are greater than b. Similarl, b is a lower bound if no real zeros of f are less than b. Upper and Lower Bound Rules Let f be a polnomial with real coefficients and a positive leading coefficient. Suppose f is divided b c, using snthetic division.. If c > and each number in the last row is either positive or zero, c is an upper bound for the real zeros of f.. If c < and the numbers in the last row are alternatel positive and negative (zero entries count as positive or negative), c is a lower bound for the real zeros of f. Eample Finding the Zeros of a Polnomial Function In Eample, notice how the Rational Zero Test, Descartes s Rule of Signs, and the Upper and Lower Bound Rules ma be used together in a search for all real zeros of a polnomial function. Find the real zeros of f 6 4. The possible real zeros are as follows. Factors of Factors of 6 ±, ± ±, ±, ±, ±6 The original polnomial f has three variations in sign. The polnomial f 6 4 has no variations in sign. As a result of these two findings, ou can appl Descartes s Rule of Signs to conclude that there are three positive real zeros or one positive real zero, and no negative zeros. Tring produces the following ±, ±, ±, ± 6, ±, ± So, is not a zero, but because the last row has all positive entries, ou know that is an upper bound for the real zeros. So, ou can restrict the search to zeros between and. B trial and error, ou can determine that is a zero. So, f 6. Because 6 has no real zeros, it follows that is the onl real zero. Now tr Eercise 87.

10 _.qd /7/ 9:6 AM Page Chapter Polnomial and Rational Functions Before concluding this section, here are two additional hints that can help ou find the real zeros of a polnomial.. If the terms of f have a common monomial factor, it should be factored out before appling the tests in this section. For instance, b writing f 4 ou can see that is a zero of f and that the remaining zeros can be obtained b analzing the cubic factor.. If ou are able to find all but two zeros of f, ou can alwas use the Quadratic Formula on the remaining quadratic factor. For instance, if ou succeeded in writing f 4 4 ou can appl the Quadratic Formula to 4 to conclude that the two remaining zeros are and. Eample Using a Polnomial Model Activities. Write as a product of linear factors: f 4 6. Answer: i i. Find a third-degree polnomial with integer coefficients that has, i, and i as zeros. Answer: 8. Use the zero i to find all the zeros of f Answer:,, i, i You are designing candle-making kits. Each kit contains cubic inches of candle wa and a mold for making a pramid-shaped candle. You want the height of the candle to be inches less than the length of each side of the candle s square base. What should the dimensions of our candle mold be? The volume of a pramid is V Bh, where B is the area of the base and h is the height. The area of the base is and the height is. So, the volume of the pramid is V. Substituting for the volume ields the following. Substitute for V. 7 7 Multipl each side b. Write in general form. The possible rational solutions are ±, ±, ±, ±, ±, ±7. Use snthetic division to test some of the possible solutions. Note that in this case, it makes sense to test onl positive -values. Using snthetic division, ou can determine that is a solution. 7 7 The other two solutions, which satisf, are imaginar and can be discarded. You can conclude that the base of the candle mold should be inches b inches and the height of the mold should be inches. Now tr Eercise 7.

11 _.qd /7/ 9:6 AM Page 79 Section. Zeros of Polnomial Functions 79. Eercises VOCABULARY CHECK: Fill in the blanks.. The of states that if f is a polnomial of degree n n >, then f has at least one zero in the comple number sstem.. The states that if is a polnomial of degree then has precisel linear factors f a n c c... f n n >, f n c n where c, c,..., c n are comple numbers.. The test that gives a list of the possible rational zeros of a polnomial function is called the Test. 4. If a bi is a comple zero of a polnomial with real coefficients, then so is its, a bi.. A quadratic factor that cannot be factored further as a product of linear factors containing real numbers is said to be over the. 6. The theorem that can be used to determine the possible numbers of positive real zeros and negative real zeros of a function is called of. 7. A real number b is a(n) bound for the real zeros of f if no real zeros are less than b, and is a(n) bound if no real zeros are greater than b. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at In Eercises 6, find all the zeros of the function.. f 6. f. g ) 4 4. f 8. f 6 i i 6. h t t t t i t i In Eercises 7, use the Rational Zero Test to list all possible rational zeros of f. Verif that the zeros of f shown on the graph are contained in the list. 7. f 9.. f f f In Eercises, find all the rational zeros of the function.. f 6 6. f 7 6. g h 9. h t t t t 6. p C 8. f f f 4

12 _.qd /7/ 9:6 AM Page 8 8 Chapter Polnomial and Rational Functions In Eercises 4, find all real solutions of the polnomial equation.. z 4 z z In Eercises 8, (a) list the possible rational zeros of f, (b) sketch the graph of f so that some of the possible zeros in part (a) can be disregarded, and then (c) determine all real zeros of f.. f f f f 4 In Eercises 9, (a) list the possible rational zeros of f, (b) use a graphing utilit to graph f so that some of the possible zeros in part (a) can be disregarded, and then (c) determine all real zeros of f. 9. f 4 8. f f 7. f Graphical Analsis In Eercises 6, (a) use the zero or root feature of a graphing utilit to approimate the zeros of the function accurate to three decimal places, (b) determine one of the eact zeros (use snthetic division to verif our result), and (c) factor the polnomial completel.. f 4 4. P t t 4 7t. 6. h g In Eercises 7 4, find a polnomial function with real coefficients that has the given zeros. (There are man correct answers.) 7., i, i 8. 4, i, i 9. 6, i, i 4., 4 i, 4 i 4.,, i 4.,, i In Eercises 4 46, write the polnomial (a) as the product of factors that are irreducible over the rationals, (b) as the product of linear and quadratic factors that are irreducible over the reals,and (c) in completel factored form. 4. f f 4 8 (Hint: One factor is 6. ) 4. f (Hint: One factor is. ) 46. f 4 (Hint: One factor is 4. ) In Eercises 47 4, use the given zero to find all the zeros of the function. Function 47. f f f g g 4 4. h f 4 4. f 4 4 Zero i i In Eercises 7, find all the zeros of the function and write the polnomial as a product of linear factors.. f 6. f 6 7. h 4 8. g f 4 8 f 4 6 f z z z 6. h() 4 6. g f 6. h h f g g h f f 4 9 In Eercises 7 78, find all the zeros of the function. When there is an etended list of possible rational zeros, use a graphing utilit to graph the function in order to discard an rational zeros that are obviousl not zeros of the function. 7. f f s s s s 7. f f f g i i i i i i

13 _.qd /7/ 9:6 AM Page 8 Section. Zeros of Polnomial Functions 8 In Eercises 79 86, use Descartes s Rule of Signs to determine the possible numbers of positive and negative zeros of the function. 79. g 8. h h 4 8. h g f 4 f f In Eercises 87 9, use snthetic division to verif the upper and lower bounds of the real zeros of f. 87. f 4 4 (a) Upper: 4 (b) Lower: 88. f 8 (a) Upper: 4 (b) Lower: 89. f (a) Upper: (b) Lower: 9. f 4 8 (a) Upper: (b) Lower: 4 In Eercises 9 94, find all the real zeros of the function. 9. f 4 9. f z z 4z 7z 9 9. f g In Eercises 9 98, find all the rational zeros of the polnomial function. 9. P f f f z z 6 z z 6 6z z z In Eercises 99, match the cubic function with the numbers of rational and irrational zeros. (a) Rational zeros: ; irrational zeros: (b) Rational zeros: ; irrational zeros: (c) Rational zeros: ; irrational zeros: (d) Rational zeros: ; irrational zeros: 99. f. f. f. f. Geometr An open bo is to be made from a rectangular piece of material, centimeters b 9 centimeters, b cutting equal squares from the corners and turning up the sides. (a) Let represent the length of the sides of the squares removed. Draw a diagram showing the squares removed from the original piece of material and the resulting dimensions of the open bo. (b) Use the diagram to write the volume V of the bo as a function of. Determine the domain of the function. (c) Sketch the graph of the function and approimate the dimensions of the bo that will ield a maimum volume. (d) Find values of such that V 6. Which of these values is a phsical impossibilit in the construction of the bo? Eplain. 4. Geometr A rectangular package to be sent b a deliver service (see figure) can have a maimum combined length and girth (perimeter of a cross section) of inches. (a) Show that the volume of the package is V 4. (b) Use a graphing utilit to graph the function and approimate the dimensions of the package that will ield a maimum volume. (c) Find values of such that V,. Which of these values is a phsical impossibilit in the construction of the package? Eplain.. Advertising Cost A compan that produces MP plaers estimates that the profit P (in dollars) for selling a particular model is given b P 76 48,, where is the advertising epense (in tens of thousands of dollars). Using this model, find the smaller of two advertising amounts that will ield a profit of $,,. 6. Advertising Cost A compan that manufactures biccles estimates that the profit P (in dollars) for selling a particular model is given b P 4 7,, 6 where is the advertising epense (in tens of thousands of dollars). Using this model, find the smaller of two advertising amounts that will ield a profit of $8,.

14 _.qd /7/ 9:6 AM Page 8 8 Chapter Polnomial and Rational Functions 7. Geometr A bulk food storage bin with dimensions feet b feet b 4 feet needs to be increased in size to hold five times as much food as the current bin. (Assume each dimension is increased b the same amount.) (a) Write a function that represents the volume V of the new bin. (b) Find the dimensions of the new bin. 8. Geometr A rancher wants to enlarge an eisting rectangular corral such that the total area of the new corral is. times that of the original corral. The current corral s dimensions are feet b 6 feet. The rancher wants to increase each dimension b the same amount. (a) Write a function that represents the area A of the new corral. (b) Find the dimensions of the new corral. (c) A rancher wants to add a length to the sides of the corral that are 6 feet, and twice the length to the sides that are feet, such that the total area of the new corral is. times that of the original corral. Repeat parts (a) and (b). Eplain our results. 9. Cost The ordering and transportation cost C (in thousands of dollars) for the components used in manufacturing a product is given b C, where is the order size (in hundreds). In calculus, it can be shown that the cost is a minimum when 4 4 6,. Use a calculator to approimate the optimal order size to the nearest hundred units.. Height of a Baseball A baseball is thrown upward from a height of 6 feet with an initial velocit of 48 feet per second, and its height h (in feet) is h t 6t 48t 6, t where t is the time (in seconds). You are told the ball reaches a height of 64 feet. Is this possible?. Profit The demand equation for a certain product is p 4., where p is the unit price (in dollars) of the product and is the number of units produced and sold. The cost equation for the product is C 8,, where C is the total cost (in dollars) and is the number of units produced. The total profit obtained b producing and selling units is P R C p C. You are working in the marketing department of the compan that produces this product, and ou are asked to determine a price p that will ield a profit of 9 million dollars. Is this possible? Eplain.. Athletics The attendance A (in millions) at NCAA women s college basketball games for the ears 997 through is shown in the table, where t represents the ear, with t 7 corresponding to 997. (Source: National Collegiate Athletic Association) Snthesis Year, t (a) Use the regression feature of a graphing utilit to find a cubic model for the data. (b) Use the graphing utilit to create a scatter plot of the data. Then graph the model and the scatter plot in the same viewing window. How do the compare? (c) According to the model found in part (a), in what ear did attendance reach 8. million? (d) According to the model found in part (a), in what ear did attendance reach 9 million? (e) According to the right-hand behavior of the model, will the attendance continue to increase? Eplain. True or False? In Eercises and 4, decide whether the statement is true or false. Justif our answer.. It is possible for a third-degree polnomial function with integer coefficients to have no real zeros. 4. If i is a zero of the function given b f i i Model It Attendance, A then i must also be a zero of f. Think About It In Eercises, determine (if possible) the zeros of the function g if the function f has zeros at r, r, and r.. g f 6. g f

15 _.qd /7/ 9:6 AM Page 8 Section. Zeros of Polnomial Functions 8 7. g f 8. g f 9. g f. g f. Eploration Use a graphing utilit to graph the function given b f 4 4 k for different values of k. Find values of k such that the zeros of f satisf the specified characteristics. (Some parts do not have unique answers.) (a) Four real zeros (b) Two real zeros, each of multiplicit (c) Two real zeros and two comple zeros (d) Four comple zeros. Think About It Will the answers to Eercise change for the function g? (a) g f (b) g f. Think About It A third-degree polnomial function f has real zeros,, and, and its leading coefficient is negative. Write an equation for f. Sketch the graph of f. How man different polnomial functions are possible for f? 4. Think About It Sketch the graph of a fifth-degree polnomial function whose leading coefficient is positive and that has one zero at of multiplicit.. Writing Compile a list of all the various techniques for factoring a polnomial that have been covered so far in the tet. Give an eample illustrating each technique, and write a paragraph discussing when the use of each technique is appropriate. 6. Use the information in the table to answer each question. Interval,,, 4 4, Value of f Positive Negative Negative Positive (a) What are the three real zeros of the polnomial function f? (b) What can be said about the behavior of the graph of f at? (c) What is the least possible degree of f? Eplain. Can the degree of f ever be odd? Eplain. (d) Is the leading coefficient of f positive or negative? Eplain. (e) Write an equation for f. (There are man correct answers.) (f) Sketch a graph of the equation ou wrote in part (e). 7. (a) Find a quadratic function f (with integer coefficients) that has ± bi as zeros. Assume that b is a positive integer. (b) Find a quadratic function f (with integer coefficients) that has a ± bi as zeros. Assume that b is a positive integer. 8. Graphical Reasoning The graph of one of the following functions is shown below. Identif the function shown in the graph. Eplain wh each of the others is not the correct function. Use a graphing utilit to verif our result. (a) f ). (b) g ). (c) h ). (d) k ). Skills Review In Eercises 9, perform the operation and simplif. 9. 6i 8 i. i 6i. 6 i 7i. 9 i 9 i In Eercises 8, use the graph of f to sketch the graph of g. To print an enlarged cop of the graph, go to the website g f 4. g f. g f 6. g f 7. g f 8. g f (, ) (, ) (, ) (4, 4) f 4