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# COLLEGE ALGEBRA MATH 124

Hoyt Beer

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Staff

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## Popular in Mathematics (M)

This 11 page Class Notes was uploaded by Hoyt Beer on Sunday October 25, 2015. The Class Notes belongs to MATH 124 at University of Nevada - Las Vegas taught by Staff in Fall. Since its upload, it has received 61 views. For similar materials see /class/228628/math-124-university-of-nevada-las-vegas in Mathematics (M) at University of Nevada - Las Vegas.

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Date Created: 10/25/15
Chapter R Section 4 Page 1 of 3 Section R4 Factoring Common Factors 0 Common Factors occur when there is a factor that is common to all terms in the expression 0 What are the common factors in the expression below 5x 25 8x2y2 4xy3 3xy Grouping 0 When you have two or more elements in an expression with common factors you can group them together and factor the common factors out This doesn t happen all too often in real life Does the expression below have common factors Can you factor by grouping ySy22y2 Factoring Quadratics ax2 bx c o Ifa l l The solution will be 0fthe form xiDXxiD and your goal is to nd what goes inside the boxes along with the signs in front of them 2 Look at the sign ofc if it is plus then you have the same sign both plus or both minus Keep in mind sum if it is different then you have opposite signs Keep in mind difference 3 Identify the factors of c nd the factors whose sum or difference is b place these values in the boxes be careful with the difference as the larger value should have the sign of b 0 Example Factor t2 2t 15 c 715 so the signs are different The factors of 15 are 115 and 35 We are looking for the difference of those factors to be 2 so we choose 35 Since we want the difference to be negative we choose 75 and 3 x 7 5x 3 o If a 1 Be sure that you can t factor out a constant term Well there is a method for factoring when it is in this form but it often causes frustration on the part of every student so just use the quadratic formula bilb2 4ac 2a E The formula will yield two values and you factor accordingly C Bellomo revised 26Aug07 Chapter R Section 4 Page 2 of 3 0 Example Factor 20p2 23p 6 23r 232 4206 23 7 2 220 40 3 2 x Zx or 4x 35x 2 How are these two expressions above the same 32 4 5 S gecial F actorz39zatz39 ons 0 There are some factorizations that occur more frequently 0 The difference of squares is given by A2 B2 A BA 3 0 A binomial squared is given by either A B2 A2 2AB 132 or A B2 A2 2AB 132 0 The sum of cubes is given by A3 B3 A BA2 AB 32 0 The difference of cubes is given by A3 33 A BA2 AB BZ Pulling itAll Together for P 39 v of One Variable continue in order stop when factored 0 Look for a greatest common factor and if possible factor it out o What degree is the expression 1 If it is a quadratic degree 2 How many terms are there If there are two terms in the form ax2 c with 0 negative It is the difference of squares Factor out the coefficient in front of the x2 term a The resulting factorization will be a x x a a If there are two terms in the form ax2 c with 0 positive You can t factor this in the real number system you re done If there are two terms in the form ax2 bx You should have already taken care of this Factor out an x The resulting factorization will be xax b Ifthere are three terms ax2 bx c Is a 1 If a 1 try factoring it see 39Factoring Quadratics39 If a 1 If a at 1 use the quadratic formula Keep in mind this doesn t have to factor 2 If it is a cubic How many terms are there If there are two terms and one is constant it is the sum or difference of cubes If there are two terms and one is not constant you should have already taken care of this Factor out the greatest common factor If there are three terms with no common factor you are stuck because it is beyond our scope for this class If there are 4 terms with no common factors try grouping 3 If it is of degree higher than 3 and there are no common factors We are extending a bit beyond our skill set unless it is of the form presented in l or 2 In other words m4 m2 90 is quadratic in form and you must think of it as quot122 m2 90 C Bellomo revised 26Aug07 Chapter R Section 4 Page 3 of 3 Remember you are not nished factoring just because you complete one step Be sure it is factored all the way Example Factor p3 2p2 9p l 8 It has no common factors It is of degree 3 cubic and has 4 terms so we will try grouping P2P 2 9P 2 P2 9P 2 Now we are not done since we still have a quadratic inside The quadratic has two terms and is the difference of squares Solution p 3p 3p 2 Example Factor 2lx2y 2xy 8y Common factor of y y2 1x2 2x 8 2 i l22 42 l 8 2 i 26 Quadratic with three terms use the quadratic formula 12 221 42 7 3 Solution y7x 43x 2 Example Factor 125a 8a4 Common factor of a al25 8a3 It is a cubic with two terms and one is a constant so it is the difference of cubes We use the formula A3 33 A BA2 AB BZ Solution a53 2a3 a5 2a52 52a 2a2 a5 2a25 10a 4a2 Now we have the last piece which is a quadratic with three terms a not equal to l 10i1l 102 44 25 ix Use the quadratic formula 21 X 10 8 300 Solution a5 2a25 10a 4a2 No real roots C Bellomo revised 26Aug07 Chapter 4 Section 5 Page 1 of 1 Section 45 Solving Exponential and Log Equations Solving Exponential Equations Equations with variables in the exponent are called exponential equations If the base is the same on both sides of the equation you can just equate the exponents Example Solve 2 32 2x 25 3 x 5 l 3x24x Example Solve 3x24x 373 x2 4x3 0 x 3 l Trouble comes when the base is not the same You will then have to solve the equation using logs Example Solve 2 40 ln2 ln40 In 40 ln2 Example Solve 250 187x 0 250 187 In 250 x lnl87 Example Solve ex 6e39x l e ex 6 0 e 3e 2 0 ex 3 xln3 x e 2 never Solving Log Equations Often it is useful to change to exponential in form Example Solve log2 x 3 2393 x 3 x 18 Example Solve log5 8 7x 3 53 8 7x 7x ll7 3 x ll77 Example Solve log x logx 3 1 log 1 x3 x x3 0lx3x 3 09x03 3 xl3 10 1 C Bellomo revised 24Oct07 Chapter 1 Section 5 Page 1 of 2 Section 15 7 Y 39 Depremino and Constant Functions What happens when you compare successive values of x a If a function rises from left to right it is said to be increasing over that domain a If a function drops from left to right it is said to be decreasing over that domain a If a function stays the same from left to right it is said to be constant over that domain a On each of the domains given is the following function increasing decreasing constant or neither M b C C d 18 81 a c l l l I 4 l l l l I l l C e a b c d e f b e Formally a function is increasing on the open interval I if for all a and b in I with a lt b a lt b Formally a function is decreasing on the open interval I if for all a and b in I with a lt b a gt b Formally a function is constant on the open interval I if for all a and b in I with a lt b a b NOTE Increasing and decreasing mean strictly there is no equality allowed Relative Maxima and Minima Relative max and min are poinm where the function achieves its highest or lowest value LOCALLY This means it is big or small compared to the points around it It need not be the biggest or smallest overall Formally for c in the domain of f c is a relative maximum of x if there exists an I such that c gt x for all x in I where x it c Formally for c in the domain off c is a relative minimum of x if there exists anI such that c lt x for all x in I where x it 0 Example List all relative max or min points C Bellomo revised 18Sep07 Chapter 1 Section 5 Page 2 of 2 Piecewise Functions When the domain is broken up into pieces and different functions apply to different parts of the domain the function is known as a piecewise function The most popular but often misconstrued piecewise function is the absolute value function The de nition of the absolute value function is given as x x Z 0 lxl x x S 0 It is often taught in school as quotstrip away the minus if it s positivequot but this is actually incorrect If the value of x is negative you multiply x by a negative 1 Trouble occurs when you move away from numbers into symbols In other words what is the absolute value of a Example Compute lx 4 x 4 x Z 4 Ix 4 4 x x lt 4 5x 8 x lt 2 1 Example For thefunction fx Ex 5 2 S x S 4 final lO 2x xgt4 f74 754 7 8 12 f2 25 4 f4457 f6 10 7 26 72 Example For the picture givensee photo determine the domain and range and write the equation of the function The domain is given from 4 to 6 We have to nd three different line segments One line goes through the points 5 and 00 Another goes through the points 00 and 33 The last goes through the points 3 2 and 6 72 x 6 S x S 0 The function is given by the equation f x x 0 lt x lt 3 2 3 S x S 6 The Greatest Integer Function The greatest integer function f x the greatest integer less than or equal to x NOTE The symbol given here is not quite right because of my text capabilities on the computer Some examples quot21quot 2 quot29quot 2 01 C Bellomo revised 18Sep07 Chapter 1 Section 4 Page 1 of 1 Section 14 Equations of Lines and Modeling Recall the two forms for the equation of a line we have seen so far Slope Intercept y mx b Point Slope y iyl mx 7 xl Recall that if a line is horizontal its slope is 0 y b and if a line is vertical its slope is unde ned x a Some Example Problems Example Final the slope anal y intercept of the equation 5x 2y 9 0 y gx Slope 52 yintercept 92 Example Final the equation if m 72 and it passes through 75 l y 1 2x 51 y 2x 9 Example Final the equations of the horizontal anal vertical lines through 0030 Horizontal y 0 Vertical x 003 Parallel anal Perpendicular Lines Two lines are parallel if they have exactly the same slope Two lines are perpendicular if their slopes are negative reciprocals In other words if one slope is l m the other 1s m Example page 109 number 54 Are the lines parallel perpenalicular or neither 2x 5y 3 2x5y 4 The slope of 1 line is 25 and the slope of the other is 725 So they are neither Example page 109 number 62 Write the equation of the line through 8 72 that is parallel and perpendicular to y 42X 7 3 1 For the parallel line its slope is 42 so the equation would be y 2 42x 8 For the perpendicular line its slope is 7142 4124 so the equation is y 2 024x 8 C Bellomo revised 18Sep07 Chapter 4 Section 3 Page 1 of 2 Section 43 Log Functions and Graphs Low 0 We want to undo the exponential function by x o This is true if and only if y log X y is the exponent b is the base and x is the argument So the log function is the inverse of the exponential function What are our conditions on x and b Working with Logs Changing forms 0 Example log101000 3 103 1000 l l 0 Exam le 10 10y 2 p gm 100 102 y 0 Example log0516 y 05y 16 y 4 0 Example Final ln e395 y ey e395 y 5 0 Example Convert equot 4000 to log log 4000 t Special Log Bases 0 Log base e is natural log written ln 0 Log base 10 is common log written log 0 These will be the only two on your calculator So if you need to calculate say log 2 you have to use the change of base formula logb M M loga b 10 2111 210g2 05 ln 4 log 4 0 Example Final log5 3 1700 In 1700 10 1700 m 446 g 11153 C Bellomo revised 24Oct07 Chapter 4 Section 3 Page 2 of 2 Graphing the Log Function 0 Domain and range of yx log x i by x x gt 0 domain y e R range 0 x intercept of yx log x y03xl 10 There is no y intercept o The graph of yx log x is not de ned when x is less than or equal to 0 0 Example Graph yx In x The graph is nonexistent when x is less than or equal to 0 The graph crosses the x aXis when x l In fact recall that this is the inverse function of 6quot C Bellomo revised 24Oct07 Chapter 4 Section 4 Page 1 of 1 Section 44 Progerties of Log Functions Log Rules log I 1 log 1 0 log by p logbUWV log M log N 10gb Mp plogb M logbMN log M logb N blag 7 Sim li in L0 Ex ressions 0 Example Express 10g02x as a sum of logs 10g02x10g0210gx Example Express loga as a ali erence of logs loga loga 76 10ga 13 asb8 Example Express loga in terms of sums and differences azbs a b8 1 1 1 loga lbs loga aA39b312 Eloga a4b3 Eloga a4 loga b3 E4loga a310ga b a Example simpliij 210g5 x log5 y 310g5 z Z 210g5 x logs y 310g5 z 10g5 x2 10g5 y 1 10g5 2 3 10g5 x 3 yz Example simpliij 10gy3z2 310gxJ 210gx z 10gy322 310gx 210gxz 10gy32210gx393y393210gxzz392 32 y x 3 2 73 732 2 72 10gyzx y xz 10g Calculating Values 0 Example Final loga 9given loga 2 m 0301 loga 7 m 0845 and loga 11m 1041 log 9 loga 7 2 but we don t have a formula to break that down We could nd log 22 loga 211 loga 210g011 03011041 1342 0 Example Given 10gb 5 m 1609 and 10gb 3 m1099final 10gb 15b 10gb 15b 10gb 310gb 5 logb b 109916091 3708 C Bellomo revised 24Oct07

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