CHAPTER 4 S TUDY G UIDE AND A SSESSMENT - Weebly

CHAPTERSTUDY GUIDE AND ASSESSMENT4VOCABULARYcompleting the square(p. 213)complex number (p. 206)conjugate (p. 216)degree (p. 206)depressed polynomial(p. 224)Descartes’ Rule of Signs(p. 231)discriminant (p. 215)extraneous solution (p. 251)Factor Theorem (p. 224)Fundamental Theorem ofAlgebra (p. 207)imaginary number (p. 206)radical equation (p. 251)radical inequality (p. 253)rational equation (p. 243)rational inequality (p. 245)Rational Root Theorem(p. 229)Remainder Theorem (p. 222)root (p. 206)synthetic division (p. 223)upper bound (p. 238)Upper Bound Theorem(p. 238)zero (p. 206)Integral Root Theorem(p. 230)leading coefficient (p. 206)Location Principle (p. 236)lower bound (p. 238)Lower Bound Theorem(p. 238)partial fractions (p. 244)polynomial equation (p. 206)polynomial function (p. 206)polynomial in one variable(p. 205)pure imaginary number(p. 206)Quadratic Formula (p. 215)UNDERSTANDING AND USING THE VOCABULARYChoose the correct term from the list to complete each sentence.1. The?2. The?can be used to solve any quadratic equation.states that if the leading coefficient of a polynomialequation a0 has a value of 1, then any rational root must be factorsof an.3. For a rational equation, any possible solution that results with a?in the denominator must be excluded from the list of solutions.?states that the binomial x r is a factor of the polynomialP(x) if and only if P(r) 0.4. The5. Descartes’ Rule of Signs can be used to determine the possible numberof positive real zeros a?has.?6. A(n)of the zeros of P(x) can be found by determining an upperbound for the zeros of P( x).?7.solutions do not satisfy the original equation.?8. Since the x-axis only represents real numbers,of a polynomialcomplex numberscomplex rootsdiscriminantextraneousFactor TheoremIntegral RootTheoremlower boundpolynomial functionquadratic equationQuadratic Formularadical equationsynthetic divisionzerofunction cannot be determined by using a graph.9. The Fundamental Theorem of Algebra states that every polynomial equation with degree greaterthan zero has at least one root in the set of10. A?is a special polynomial equation with a degree of two.For additional review and practice for each lesson, visit: www.amc.glencoe.comChapter 4 Study Guide and Assessment267

CHAPTER 4 STUDY GUIDE AND ASSESSMENTSKILLS AND CONCEPTSOBJECTIVES AND EXAMPLESLesson 4-1Determine roots of polynomialequations.Determine whether each number is a root ofa3 3a2 3a 4 0. Explain.13. 2Determine whether 2 is a root ofx 4 3x3 x2 x 0. Explain.11. 0f(2) 24 3(23) 22 2f(2) 16 24 4 2 or 14Since f(2) 0, 2 is not a root ofx4 3x3 x2 x 0.14. Is 3 a root of t 4 2t 2 3t 1 0?Lesson 4-2Solve quadratic equations.Find the discriminant of 3x2 2x 5 0and describe the nature of the roots of theequation. Then solve the equation by usingthe Quadratic Formula.The value of the discriminant, b2 4ac, is( 2)2 4(3)( 5) or 64. Since the value ofthe discriminant is greater than zero, thereare two distinct real roots. x bb2 4ac2a ( 2) 64 2(3)2 8x 65x 1 or 3x Lesson 4-3 Find the factors of polynomialsusing the Remainder and Factor Theorems.Use the Remainder Theorem to find theremainder when (x3 2x2 5x 9) isdivided by (x 3). State whether thebinomial is a factor of the polynomial.f(x) x3 2x2 5x 9f( 3) ( 3)3 2( 3)2 5( 3) 9 27 18 15 9 or 3Since f( 3) 3, the remainder is 3. Sothe binomial x 3 is not a factor of thepolynomial by the Remainder Theorem.268REVIEW EXERCISESChapter 4 Polynomial and Rational Functions12. 415. State the number of complex roots of theequation x 3 2x2 3x 0. Then findthe roots and graph the related function.Find the discriminant of each equation anddescribe the nature of the roots of theequation. Then solve the equation by usingthe Quadratic Formula.16. 2x2 7x 4 017. 3m2 10m 5 018. x2 x 6 019. 2y2 3y 8 020. a2 4a 4 021. 5r 2 r 10 0Use the Remainder Theorem to find theremainder for each division. State whether thebinomial is a factor of the polynomial.22. (x3 x2 10x 8)23. (2x3 5x2 7x 1)24. (4x3 7x 1)25. (x4 10x2 9) (x 2)(x 5)1x 2 (x 3)

CHAPTER 4 STUDY GUIDE AND ASSESSMENTOBJECTIVES AND EXAMPLESREVIEW EXERCISESLesson 4-4List the possible rational roots of eachequation. Then determine the rationalroots.Identify all possible rational rootsof a polynomial equation by using the RationalRoot Theorem.26. x3 2x2 x 2 0List the possible rational roots of4x3 x2 x 5 0. Then determine therational roots.27. x4 x2 x 1 028. 2x3 2x2 2x 4 0pqIf is a root of the equation, then p is a29. 2x4 3x3 6x2 11x 3 0factor of 5 and q is a factor of 4.possible values of p: 1, 5possible values of q: 1, 2, 4possible rational roots:1 ,45 ,21,5,5 430. x5 7x3 x2 12x 4 031. 3x3 7x2 2x 8 01 ,232. 4x3 x2 8x 2 033. x4 4x2 5 054Graphing and substitution show a zero at .Lesson 4-4 Determine the number and type ofreal roots a polynomial function has.For f(x) 3x 4 9x 3 4x 6, there arethree sign changes. So there are three orone positive real zeros.For f( x) 3x 4 9x 3 4x 6, there isone sign change. So there is one negativereal zero.Lesson 4-5 Approximate the real zeros of apolynomial function.Determine between which consecutiveintegers the real zeros of f(x) x 3 4x 2 x 2 are located.Use synthetic division.r14 4 101 6 3 1124 2 12 34 1 13 200 141 21 156434. f(x) x 3 x 2 34x 5635. f(x) 2x 3 11x 2 12x 936. f(x) x 4 13x 2 36Determine between which consecutive integersthe real zeros of each function are located.37. g(x) 3x 3 138. f(x) x 2 4x 239. g(x) x 2 3x 340. f(x) x3 x 2 1 21Find the number of possible positive real zerosand the number of possible negative realzeros for each function. Then determinethe rational zeros.changein signs41. g(x) 4x 3 x 2 11x 342. f(x) 9x 3 25x 2 24x 643. Approximate the real zeros ofzerochangein signsf(x) 2x 3 9x 2 12x 40 to thenearest tenth.One zero is 1. Another is located between 4 and 3. The other is between 0 and 1.Chapter 4 Study Guide and Assessment269

CHAPTER 4 STUDY GUIDE AND ASSESSMENTOBJECTIVES AND EXAMPLESLesson 4-6Solve rational equations and111Solve 2 .92aa111 92aa2111 (18a2 ) (18a 2 )92aa2 Solve each equation or inequality.644. n 5 0n1x 345. x2x 22m1546. 2m 23m 3635 47. 2yy2148. 1 x 1x 1inequalities. REVIEW EXERCISES 2a 2 9a 182a 2 9a 18 0(2a 3)(a 6) 032a or 6Lesson 4-7Solve radical equations andinequalities.Solve each equation or inequality.49. 5 x 2 0Solve 9 x 1 1.9 x 1 1x 1 8x 1 64x 6550. 4a 1 8 5351. 3 x 8 x 3552. x 5753. 4 2a 7 6Lesson 4-8Write polynomial functions tomodel real-world data.54. Determine the type of polynomial functionthat would best fit the scatter plot.Determine the type of polynomial functionthat would best fit the data in the scatterplot.yyxO55. Write a polynomial function to model theOxThe scatter plot seems to change directionthree times. So a quartic function wouldbest fit the scatter plot.270Chapter 4 Polynomial and Rational Functionsdata.x 3 10247f(x)246393194

CHAPTER 4 STUDY GUIDE AND ASSESSMENTAPPLICATIONS AND PROBLEM SOLVING56. EntertainmentThe scenery for a newchildren’s show has a playhouse with apainted window. A special gloss paint coversthe area of the window to make them looklike glass. If the gloss only covers 315 squareinches and the window must be 6 inchestaller than it is wide, how large shouldthe scenery painters make the window?(Lesson 4-1)57. Gardening The length of a rectangularflower garden is 6 feet more than its width. Awalkway 3 feet wide surrounds the outsideof the garden. The total area of the walkwayitself is 288 square feet. Find the dimensionsof the garden. (Lesson 4-2)58. MedicineDoctors can measure cardiacoutput in potential heart attack patients bymonitoring the concentration of dye after aknown amount in injected in a vein near theheart. In a normal heart, the concentrationof the dye is given by g(x) 0.006x 4 0.140x 3 0.053x 2 1.79x, where x is thetime in seconds. (Lesson 4-4)a. Graph g(x)b. Find all the zeros of this function.59. PhysicsThe formula T 2 g is used tofind the period T of a oscillating pendulum.In this formula, is the length of thependulum, and g is acceleration due togravity. Acceleration due to gravity is9.8 meters per second squared. If apendulum has an oscillation period of1.6 seconds, determine the length of thependulum. (Lesson 4-7)ALTERNATIVE ASSESSMENTOPEN-ENDED ASSESSMENTb. Solve your equation. Identify theextraneous solution and explain why it isextraneous.3. a. Write a set of data that is bestrepresented by a cubic equation.b. Write a polynomial function to model theset of data.c. Approximate the real zeros of thepolynomial function to the nearest tenth.PORTFOLIOExplain how you can use the leadingcoefficient and the degree of a polynomialequation to determine the number of possibleroots of the equation.ProjectEBEDtwo solutions, one which is 2. Solve yourequation.2. a. Write a radical equation that hassolutions of 3 and 6, one of which isextraneous.LDUnit 1WI1. Write a rational equation that has at leastWWTELECOMMUNICATIONThe Pen is Mightier than the Sword! Gather all materials obtained from yourresearch for the mini-projects in Chapters 1,2, and 3. Decide what types of software wouldhelp you to prepare a presentation. Research websites that offer downloads ofsoftware including work processing, graphics,spreadsheet, and presentation software.Determine whether the software is ademonstration version or free shareware.Select at least two different programs for eachof the four categories listed above. Prepare a presentation of your Unit 1 projectusing the software that you found. Be surethat you include graphs and maps in thepresentation.Additional Assessmentpractice test.See p. A59 for Chapter 4Chapter 4 Study Guide and Assessment271

SAT & ACT Preparation4CHAPTERCoordinate Geometry ProblemsTEST-TAKING TIPThe ACT test usually includes several coordinate geometry problems.You’ll need to know and apply these formulas for points (x1, y1) and(x2, y2):Midpointx1 x2 y1 y2 , 22 DistanceSlope2(x2 x (y2 y1)2 1) y2 y1 x2 x1 Draw diagrams. Locate points on the grid ornumber line. Eliminate any choices thatare clearly incorrect.The SAT test includes problems that involve coordinate points. Butthey aren’t easy!ACT EXAMPLESAT EXAMPLE1. Point B(4,3) is the midpoint of line segmentAC. If point A has coordinates (0,1), thenwhat are the coordinates of point C?A ( 4, 1)B (4, 1)D (8, 5)E (8, 9)2. What is the area of square ABCD in squareunits?yA (0, 5)B(5, 4)C (4, 4)OD ( 1, 0)HINTDraw a diagram. You may be able tosolve the problem without calculations.SolutionDraw a diagram showing the knownquantities and the unknown point C.A 25B 18 2 D 25 2 E 36HINTyC (x, y)B (4, 3)A (0, 1)xOSince C lies to the right of B and the x-coordinateof A is not 4, any points with an x-coordinate of 4or less can be eliminated. So eliminate choicesA, B, and C.Use the Midpoint Formula. Consider thex-coordinates. Write an equation in x.0 x 42x 8Do the same for y.1 y 32y 5The coordinates of C are (8, 5) The answer ischoice D.272Chapter 4Polynomial and Rational FunctionsxC(4, 1)C 26Estimate the answer to eliminateimpossible choices and to check yourcalculations.SolutionFirst estimate the area. Since thesquare’s side is a little more than 5, the area is alittle more than 25. Eliminate choices A and E.To find the area, find the measure of a side andsquare it. Choose side AD, because points Aand D have simple coordinates. Use the DistanceFormula.(A D )2 ( 1 0)2 (0 5)2 ( 1)2 ( 5)2 1 25 or 26The answer is choice C.Alternate Solution You could also use thePythagorean Theorem. Draw right triangle DOA,with the right angle at O, the origin. Then DO is1, OA is 5, and DA is 26. So the area is26 square units.