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Testing CenterStudent Success CenterAccuplacer College Level Math Study GuideThe following sample questions are similar to the format and content of questions on the Accuplacer College LevelMath test. Reviewing these samples will give you a good idea of how the test works and just what mathematicaltopics you may wish to review before taking the test itself. Our purposes in providing you with this information are toaid your memory and to help you do your best.I.Factoring and expanding polynomialsFactor the following polynomials:Expand the following:1. 15a b 45a b 60a b9. x 1 x 1 x 3 2. 7 x3 y3 21x2 y 2 10 x3 y 2 30 x2 y10. 2x 3 y 3. 6 x 4 y 4 6 x3 y 2 8xy 2 811. x4. 2 x 2 7 xy 6 y 212. xy4 y2 613.6. 7 x3 56 y 314.3 25.2 322 3 3 x 6 62 2 x 3 2 x 1 6 x 1 57. 81r 4 16s 48. x y 2 2 x y 1II.Simplification of Rational Algebraic ExpressionsSimplify the following. Assume all variables are larger than zero.32 5 4 402. 9 3 5 8 2 27813.x41.III.4.2 18 5 32 7 1625.6 x 18 12 x 16 3x 2 2 x 8 4 x 123.y y 2 y 2 6Solving EquationsA. Solving Linear Equations1. 3 2 x 1 x 102.x x 12 74. 2 x 1 3x 3 x 1 B. Solving Quadratic & Polynomial Equations8 2 y 03 3 322. 2 x 4 x 30 x 033. 27 x 11. y 4.5.t 2 t 1 03x3 24227. x 1 x 256. x 3 x 6 9 x 228.5 y2 y 11August 2012College Level Math

C. Solving Rational Equations1121 x 25 x 5 x 51 6 25.a a 5 11x 6.2x 3x x x 312 0y 1 y 12312 22.x 3 x 3 x 9125x 23.6 x x 3 x 3x 184.1.D.2Solving Absolute Value Equations1.5 2z 1 84.2.x 5 7 25.3.5 x 1 213 1x 24 4y 1 7 yE. Solving Exponential Equations1.10x 10004.3x 9 x 5.2 x 42 x 2103 x 5 1001x 13. 2 8132.2F. Solving Logarithmic Equations1. log 2 x 5 log 2 1 5 x 184. ln x ln 2 x 1 02. 2log3 x 1 log3 4 x 5. ln x ln x 2 ln 33. log 2 x 1 log 2 x 1 36.32 x 4x 1G. Solving Radical Equations1. 4 2 y 1 2 02.2x 1 5 83.5x 1 2 x 1 0x2 9 x 1 04.5.33x 2 4 66.4w2 7 2IV. Solving InequalitiesSolve the following inequalities and express the answer graphically and using interval notation.A. Solving Linear Inequalities3x 4 252. 3 x 3 5 x 1 3. 3 x 2 6 x 3 141.4.2 3x 10 52October 2012College Level Math

B. Solving Absolute value Inequalities: Solve and Graph.1.4x 1 62.4x 3 2 9x 5 534. 5 2 x 153.C. Solving Quadratic or Rational InequalitiesV.1.3x2 11x 4 02.6 x2 5x 4x 2 03 x x 1 x 3 04.2x 73.Lines & Regions1. Find the x and y-intercepts, the slope, and graph 6x 5y 30.2. Find the x and y-intercepts, the slope, and graph x 3.3. Find the x and y-intercepts, the slope, and graph y -4.4. Write in slope-intercept form the line that passes through the points (4, 6) and (-4, 2).5. Write in slope-intercept form the line perpendicular to the graph of 4x - y -1 and containing the point (2, 3).6. Graph the solution set of x - y 2.7. Graph the solution set of -x 3y -6.VI. Graphing Relations, Domain & RangeFor each relation, state if it is a function, state the domain & range, and graph it.1.y x 2y x 2x 13. y x 24. f x x 1 32.5.f x 2x 5x2 96.x y2 27.y x2 8x 68.y x9.y 3x10.h x 6 x23x 2 2 x 1VII. Exponents and RadicalsSimplify. Assume all variables are 0. Rationalize the denominators when needed.1.2.3 8x35 147 4 4853. 15 3 23 43x y4. 5 x 3 5. 54a 6b 2 6. 3 8 9a b 33 7. 8.3 227 a 32a 2b 225 339.xx 340x 4y93October 2012College Level Math

VIII. Complex NumbersPerform the indicated operation and simplify.1. 1 6 4 95.2. 16 96.2i 253 2i7.4 5i 16 94. 4 3i 4 3i 3.IX. 4 3i Exponential Functions and Logarithmsx1. Graph: f x 3 16.Solve:2. Graph: g x 27.Graph: h x log3 x8.Use the properties of logarithms to expand asx 11in logarithmic form643. Express8 2 4. Expresslog5 25 2 in exponential form5. Solve:log x 9 2much as possible:log 2 x 49.log 43yHow long will it take 850 to be worth 1000 ifit is invested at 12% interest compoundedquarterly?X.Systems of Equations & Matrices2x 3y 71. Solve the system:6x y 1 1 1 1 0 2 1 4. Multiply: 02 0 1 2 0 2 1 3 0 0 1 x 2 y 2z 32. Solve the system: 2 x 3 y 6 z 2 x y z 05.Find the determinant:3. Perform the indicated operation:6.Find the Inverse:1 23 1 1 2 1 2 3 1 13 2 2 3 1 2 1 6 XI.Story Problems1. Sam made 10 more than twice what Pete earned in one month. If together they earned 760, how muchdid each earn that month?2. A woman burns up three times as many calories running as she does when walking the same distance. Ifshe runs 2 miles and walks 5 miles to burn up a total of 770 calories, how many calories does she burn upwhile running 1 mile?3. A pole is standing in a small lake. If one-sixth of thelength of the pole is in the sand at the bottom of theWater Linelake, 25 ft. are in the water, and two-thirds of the totallength is in the air above the water, what is the lengthSandof the pole?4October 2012College Level Math

XII. Conic Sections1. Graph the following, and find the center,foci, and asymptotes if possible.a)2.( x 2)2 y 2 16( x 1)2 ( y 2) 2 1b)169( x 1)2 ( y 2) 2c) 11692d) ( x 2) y 4Identify the conic section and put it into standardform.a)x2 4 x 12 y 2 0b)9 x2 18x 16 y 2 64 y 71c)9 x2 18x 16 y 2 64 y 199d)x2 y 4 x 0XIII. Sequence and Seriesan 3n 21.Write out the first four terms of the sequence whose general term is2.Write out the first four terms of the sequence whose general term is an n 13.Write out the first four terms of the sequence whose general term is an 2 14.Find the general term for the following sequence: 2,5,8,11,14,17.5.Find the general term for the following sequence:6.Find the sum:2n4, 2,1, 12 , 14 ,.6 2k 1k 047.Expand the following: 4 k xk 0 ky 4 kXIV. FunctionsLet1.2.3.4.XV.f ( x) 2 x 9 and g ( x) 16 x 2 . Find the following.5. ( g f )( 2)f ( 3) g (2)6. f ( g ( x))f (5) g (4)f ( 1) g ( 2)f (5)g (5)7.f 1 (2)8.f f 1 (3) Fundamental Counting Rule, Factorials, Permutations, & Combinations8!3! 8 3 !1.Evaluate:2.A particular new car model is available with five choices of color, three choices of transmission, four typesof interior, and two types of engines. How many different variations of this model car are possible?3.In a horse race, how many different finishes among the first three places are possible for a ten-horse race?4.How many ways can a three-Person subcommittee be selected from a committee of seven people? Howmany ways can a president, vice president, and secretary be chosen from a committee of seven people.5October 2012College Level Math

XVI. Trigonometryf ( x) sin x2. Graph the following through on period: g ( x) cos(2 x)1.Graph the following through on period:3.A man whose eye level is 6 feet above the ground stands 40 feet from a building. The angle of elevationfrom eye level to the top of the building is4.72 . How tall is the building?A man standing at the top of a 65m lighthouse observes two boats. Using the data given in the picture,determine the distance between the two boats.6October 2012College Level Math

AnswersI.Factoring and Expanding PolynomialsWhen factoring, there are three steps to keep in mind.1. Always factor out the Greatest Common Factor2. Factor what is left3. If there are four terms, consider factoring by grouping.Answers:15a 2b(ab 3b2 4)22. x y(7 y 10)( xy 3)1.7 x3 y 3 21x 2 y 2 10 x3 y 2 30 x 2 ySince there are 4 terms, we consider factoring by grouping.x y (7 xy 21y 10 xy 30)First, take out the Greatest Common Factor.22x y (7 xy 21y ) ( 10 xy 30) x 2 y 7 y ( xy 3) 10( xy 3) 224.2(3x3 y 2 4)( xy 2 1)(2 x 3 y)( x 2 y)5.( y 2 2)( y 2 3)3.y4 y2 6u2 u 6(u 2)(u 3)6.When you factor by grouping, be careful of the minus signbetween the two middle terms.When a problem looks slightly odd, we can make it appear more natural to usby using substitution (a procedure needed for calculus). Letu y2Factor2the expression with u’s. Then, substitute the y back in place of the u’s. Ifyou can factor more, proceed. Otherwise, you are done.7( x 2 y)( x2 2 xy 4 y 2 )Formula for factoring the sum of two cubes:a3 b3 (a b)(a 2 ab b2 )The difference of two cubes is:a3 b3 (a b)(a 2 ab b2 )(3r 2s)(3r 2s)(9r 2 4s 2 )28. ( x y 1)Hint: Let u x y329. x 3x x 32210. 4 x 12 xy 9 y7.3x 2 2 3 243212. x 4 x 10 x 12 x 9543213. x 5x 10 x 10 x 5x 16543214. x 6 x 15x 20 x 15x 6 x 111.When doing problems 13 and 14, you may want to usePascal’s Triangle11 11 2 11 3 3 11 4 6 4 17October 2012College Level Math

II.Simplification of Rational Algebraic Expressions1. 132. 383.4.5.III.If you have9x249 26x 24 , you can write 4 as a product of primes (2 2) . In squareroots, it takes two of the same thing on the inside to get one thing on theoutside:4 2 2 2Solving EquationsA. Solving Linear Equations1.x 52.x 144or 2553.y 34.x 1B. Solving Quadratic & Polynomial Equations1.2.3.4.8 2y , 3 3x 0, 3,51 1 i 3x , 3 66x 10, 4 1 i 35. t 226. x 2, 1 i 37. x 3, 48.y Solving Quadratics or Polynomials:1. Try to factor2. If factoring is not possible, use the quadratic formulax Note: b b 2 4ac2awhereax2 bx c 0i 1 12 i 12 i 2 2 3 2i 3Example:1 2110C. Solving Rational Equations1.y 1312 0y 1 y 1 12 ( y 1)( y 1) 0( y 1)( y 1) y 1 y 1 12( y 1)( y 1) ( y 1)( y 1) 0y 1y 1( y 1) 2( y 1) 03 y 1 02.Solving Rational Equations:1. Find the lowest common denominator forall fractions in the equation2. Multiply both sides of the equation by thelowest common denominator3. Simplify and solve for the given variable4. Check answers to make sure that theydo not cause zero to occur in thedenominators of the original equation.Working the problem, we get x 3 . However, 3 causes the denominators to be zero in the originalequation. Hence, this problem has no solution.8October 2012College Level Math

3.4.5.6.D.1.2.3.4.5.E.1.x 154x 2a 1, 5x 2,1Solving Absolute Value Equationsz 2 or z 75 2z 1 85 2z 95 2 z 9 or 5 2 z 9 2 z 4 or 2 z 14x 0 or x 10x 2means that the object insidethe absolute value has a distance of 2 away from zero.The only numbers with a distance of 2 away from zeroare 2 and -2. Hence, x 2 or x 2 . Use thesame thought process for solving other absolute valueequationsy 3y 1 7 yy 1 7 y or y 1 (7 y)or 2 y 60 8No solutionor y 3Hence, y 3 is the only solutionNote: An absolute value cannot equal a negative value.x 2does not make any sense.Note: Always check your answers!!Solving Exponential Equationsx 310 x 10005.10 10 x 3x 1x 4x 1, 1x 1, 3F.Solving Logarithmic Equations1.x 2.2. Remember thatNo solution. An absolute valuecannot equal a negative numberx 2 or x 1x2.3.4.Solving Absolute Value Equations:1. Isolate the Absolute value on one side of the equationand everything else on the other side323log 2 ( x 5) log 2 (1 5 x)x 5 1 5x6 x 4x 1,12 log 3 ( x 1) log 3 (4 x)Some properties you will need to be familiar with.IfIfa r a s , then r sa r br , then a bProperties of logarithms to be familiar with: Iflogb M logb N , then M N Iflogb x y , then this equation can be rewritten inby xlogb (M N ) logb M logb Nexponential form as log 3 ( x 1) 2 log 3 (4 x) ( x 1) 2 4 x x 2x 1 02 M logb logb M logb N N logb M r r logb MAlways check your answer!! Bases and arguments oflogarithms cannot be negative9October 2012College Level Math

3.4.5.6.x 3 is the only solution since -3 causes the argument of a logarithm to be negative1x is the only solution since -1 causes the argument of a logarithm to be negative2x 1 is the only solution since -3 causes the argument of a logarithm to be negativeln 4x 2ln 3 ln 432 x 4 x 12 x ln 3 ( x 1) ln 42 x ln 3 x ln 4 ln 4x(2 ln 3 ln 4) ln 4G. Solving Radical Equations1.y 584 2 y 1 2 02 y 1 2.3.x 4x 512142 y 1 2y Solving Equations with radicals:1. Isolate the radical on one side of the equation and everythingelse on the other side2. If it is a square root, then square both sides. If it is a cube root,then cube both sides, etc 3. Solve for the given variable and check your answerNote: A radical with an even index such as,4 ,6, cannot have a negative argument (The square root can butyou must use complex numbers).545x 1 2 x 1 05x 1 2 x 1 5x 1 22x 1 25 x 1 4 x 1 4. No solution. x 4 does not work in the original equation.5. x 26. w 3, 3IV.Solving InequalitiesA. LinearWhen solving linear inequalities, you use the same steps assolving an equation. The difference is when you multiply ordivide both sides by a negative number, you must changethe direction of the inequality.3x 4 21.53x 65x 10Interval Notation:5 3For example: 1 5 1 3 5 3 , 10 10October 2012College Level Math

2. x 7Interval Notation: , 7 3. x 4Internal Notation: 4, 4. 4 x 5Internal Notation: 4,5 B. Absolute Value1.4x 1 6Think of the inequality sign as an alligator. If the alligator is 6 4 x 1 6 7 4 x 575 x 44facing away from the absolute value sign such as,x 5,then one can remove the absolute value and write 5 x 5 . This expression indicates that x cannot befarther than 5 units away from zero.If the alligator faces the absolute value such as,x 5,then one can remove the absolute value and writex 5or x 5 . These expressions express that x cannot beless than 5 units away from zero.Interval:2.x 1 or x Interval:3.525 , 1, 2 x 20 or x 10Interval:4. 7 5 4 , 4 , 20 10, 5 x 10Interval: 5,10 11October 2012College Level Math

C. Quadratic or Rational1.3x 2 11x 4 0 3x 1 x 4 0x 1and x 4 make the above factors zero.3-3x 1x-4 --4Answer:V. 1 , 4 x 3 y 6 3 2.4 1 , , 3 2 3. 2,3 4.7 , 1,3 2 Steps to solving quadratic or rationalinequalities.1. Zero should be on one side of theinequalities while everything else is onthe other side.2. Factor3. Set the factors equal to zero andsolve.4. Draw a chart. You should have anumber line and lines dividing regionson the numbers that make the factorszero. Write the factors in on the side.5. In each region, pick a number andsubstitute it in for x in each factor.Record the sign in that region.6. In our example, 3x 1 is negative inthe first region when we substitute anumber such as -2 in for x. Moreover,3x 1 will be negative everywhere inthe first region. Likewise, x-4 will benegative throughout the whole firstregion. If x is a number in the firstregion, then both factors will benegative. Since a negative times anegative number is positive, x in thefirst region is not a solution. Continuewith step 5 until you find a region thatsatisfies the inequality.7. Especially with rational expression,check that your endpoints do notmake the original inequalityundefined.Lines and Regions1. x – intercept: (5,0)y – intercept: (0,6)slope:2. 65x – intercept: (3,0)y – intercept: Noneslope: None12October 2012College Level Math

3.x – intercept: Noney – intercept: (0,-4)slope: 01x 424.y 5.11y x 3426.x y 27. x 3 y 613October 2012College Level Math

VI.Graphing Relations1.y x 2Domain: 2, Range:2. 0, y x 2 0, 2, Domain:Range:3.y x 1x 2Domain: All Real Numbers except -2Range:4. ,1 1, f x x 1 3 , ,3 Domain:Range:14October 2012College Level Math

5.f x 2x 5x2 9Domain: All Real Number exceptRange: All Real Numbers6.x y2 2Domain: 2, Range:7. , y x2 8x 6Domain: , Range:8. 3 22, y x , 0 0, Domain:Range:15October 2012College Level Math

9.y 3x , 0, Domain:Range:10.h x 6 x23x 2 2 x 1Domain: All Real Numbers except:Range: 2, ,0 1 ,13VII. Exponents and Radicals1. 2x2.5 147 4 48 35 3 16 3 19 33.5 3 154.x7y45.2 x 3 5xy3 2 2 54a 6b 2 a 6b12 6 3 6 6. 3 8 36 a b 9a b 37.8.9.327a32a 2 b 2 3a32a 2 b 2 3a32a 2 b 2 34ab 3a 3 4ab 3 3 4ab 32ab2b4ab5 3 x x 3 x x 3x x 9 x 3 x 3 16October 2012College Level Math

VIII. Complex Numbers1.2. 16 4 9 4i 12i 8i 16 9 4i 3i 12i 2 12 16 4i 4i 3i 12i 2 12 4 2 9 3 9 3i 3i 3i 9i24. 4 3i 4 3i 16 9i 16 9 253.5. 4 3i 6.i 25 i i 24 i i 2 i 1 i7.3 2i 4 5i 12 23i 10i 2 12 23i 10 2 23i 4 5i 4 5i16 25i 216 2541 412 4 3i 4 3i 16 24i 9i 2 16 24i 9 7 24i12IX.12Exponential Functions and Logarithms1.f x 3x 12.g x 2 x 13.log84.52 251 264log 2 x 45.24 x16 x17October 2012College Level Math

6.x 3; -3 is not a solution because bases are not allowed to be negative7.h x log3 x8.log 49.3 log 4 3 log 4 yy r A P 1 n A Money ended withP Principle started withr Yearly interest raten Number of compounds per yeart Number of yearsntwhere 0.12 1000 850 1 4 20 0.12 1 17 4 4t4t4t 20 0.12 log log 1 4 17 20 0.12 log 4t log 1 4 17 20 log 17 t 0.12 4 log 1 4 18October 2012College Level Math

X.Systems of Equations3 1 , 2, 4 21.3.5.XI. 5 8 5 14 5Story Problems1.Let x the money Pete earns2.2 3 5 k 5 8 k 2 for k Natural numbers 55 k 4. 1 0 2 2 4 0 1 2 5 6. 1 2 1 4 2 x 10 x 7602 x 10 the money Sam earns2.x burned calories walkingx length of polePete earns 250Sam earns 5102 3x 5 x 770x 703x burned calories running3.1 2 1 4 Answer: 210 calories21x 25 x x36Answer: 150 feetXII. Conic Sections1.a) x 2 2 y 2 16Center: (2,0)Radius: 419October 2012College Level Math

b) x 1 216 y 2 c) 19Center: (-1,2)Foci:2 1 x 1 2167, 2 y 2 2 19Center: (-1,2)Foci: (-6,2), (4,2)33x 44Asymptotes:35y x 44y d) x 2 2 y 4Vertex: (2,4)Foci: 15 2, 4 Directrix:2.a) Circleb) Ellipsec) Hyperbolad) Parabolay 174 x 2 x 1 2 y 2 162 y 2 16 x 1 16292 y 2 9 12 1y x 2 4220October 2012College Level Math

XIII. Sequence and Series1. 1, 4, 7, 102.0, 3, 8, 153.3, 5, 9, 174.an 3n 15. 1 an 4 2 6. 2k 1 1 1 3 5 7 9 11 35n 1 2 3 n 6k 047. 4 k xk 0k y 4 k y 4 4 xy 3 6 x 2 y 2 4 x3 y x 4XIV. Functions1.f 3 g 2 3 12 152.f 5 g 4 19 0 193.f 1 g 2 7 12 844.f 5 1919 g 5 995. g6.f g x f 16 x 2 2 16 x 2 9 2 x 2 417.f 1 x 8.f 2 g f 2 g 5 9x 9;2f 1 2 2 97 22321October 2012College Level Math

XV.Fundamental Counting Rule, Permutations, & Combinations1. 562.1203.7204.Committee 35Elected 210XVI. Trigonometry1.f x sin x2.g x cos 2 x 3.x 6 40 tan 72 129.14.Distance between the boats 65tan 42 65tan 35 50 11.59meters22October 2012College Level Math

August 2012 College Level Math Accuplacer College Level Math Study Guide The following sample questions are similar to the format and content of questions on the Accuplacer College Level Math test. Reviewing these samples will give you a g