Quotient Rule: am a n = am n (a 0) Negative Exponents: a n = 1 (a 0) an Power Rules: (a m ) n = a m n (ab) m = a m b m


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1 Formuls nd Concepts MAT 099: Intermedite Algebr repring for Tests: The formuls nd concepts here m not be inclusive. You should first tke our prctice test with no notes or help to see wht mteril ou re comfortble with, nd wht mteril ou m need to review more. Once ou hve n ide of wht mteril ou need to review more, go bck to those sections to stud. repring for the Finl Em: Stud the individul prctice test (rctice Test 1, rctice Test 2, etc.), nd spend time on the sections in which ou hd the lowest scores. After ou hve prepred thoroughl then ou cn tke the rctice Finl Em. Keep in mind tht the rctice Finl Em selects 40 questions t rndom from our previous prctice tests, so it will not necessril cover ll of the mteril tht is on the finl. Things to keep in mind: Studing for the tests in this course is process tht will require firl significnt mount of time, nd should be completed over the course of few ds. Set side stud time with fellow clssmtes ech d in the ds leding up to our ems. If ou re unstisfied with our test scores, ou m need to tr different technique for studing. You should contct our instructor to discuss our stud hbits. 1. TEST 1 MATERIAL Order of opertions Simplif epressions using the order tht follows. If grouping smbols such s prentheses re present, simplif epressions with those first, strting with the innermost set. If frction brs re present, simplif the numertor nd the denomintor seprtel. (1) Evlute eponentil epressions, roots, or bsolute vlues in order from left to right. (2) Multipl or divide in order from left to right. (3) Add or subtrct in order from left to right. Algebric roperties Commuttive: + b = b + b = b Associtive: ( + b) + c = + (b + c) ( b) c = (b c) Distributive: (b + c) = b + c Solving liner equtions in one vrible (1) Cler the eqution of frctions b multipling both sides of the eqution b the lest common multiple (2) Simplif epressions in prenthesis (3) Simplif b combining like terms (4) Move the vrible terms to one side nd numbers to the other using the ddition propert of equlit (5) Isolte the vrible using the multipliction propert (divide both sides b the coefficient of the vrible) (6) Check our nswer b substituting into the originl eqution. Steps for problem solving Generl Strteg for roblem Solving (1) UNDERSTAND the problem. During this step, become comfortble with the problem. Some ws of doing this re: Red nd rered the problem. ropose solution nd check. creful ttention to how ou check our proposed solution. This will help when writing n eqution to model the problem. Construct drwing Choose vrible to represent the unknown. (Ver importnt prt) (2) TRANSLATE the problem into n eqution. (3) SOLVE the eqution. (4) INTERRET the results; Check the proposed solution in the stted problem nd stte our conclusion. 2. TEST 2 MATERIAL A liner inequlit in one vrible is n inequlit tht cn be written in the form + b < c, where, b, nd c re rel numbers nd 0. ( The inequlit smbols, >, nd lso ppl here.) Solving liner inequlit in one vrible: 1
2 2 (1) Cler the eqution of frctions. (2) Remove grouping smbols such s prentheses (3) Simplif b combining like terms. (4) Write vrible terms on one side nd numbers on the other side using the ddition propert of inequlit. (5) Isolte the vrible b dividing both sides b the coefficient of the vrible. (Note: if the coefficient is negtive, ou must lso flip the inequlit sign) A reltion is set of ordered pirs. (1) The domin of the reltion is the set of ll first components of the ordered pirs. (2) The rnge of the reltion is the set of ll second components of the ordered pirs. (3) A function is reltion in which ech first component in the ordered pirs corresponds to ectl one second component. (4) Verticl Line Test: A reltion is function if no verticl line cn be drwn which intersects the function t two distinct points. Function Nottion To denote tht is function of, we cn write = f() (Red f of ) This nottion mens tht is function of or tht depends on. For this reson, is clled the dependent vrible nd the independent vrible. 3. TEST 3 MATERIAL The slope (m) of line pssing through points ( 1, 1 ) nd ( 2, 2 ) is given b the following: m = A liner function cn be written in the following ws: Stndrd Form: A + B = C, where A, B, nd C re rel numbers. SlopeIntercept Form: = m + b, where m is the slope, nd (0, b) is the intercept. ointslope Form: 1 = m( 1 ), where m is the slope, nd ( 1, 1 ) is some point on the line. Horizontl nd Verticl Lines: = c Horizontl Line The slope is 0, nd the intercept is (0, c) = c Verticl Line The slope is undefined, nd the intercept is (c, 0) The intercept of line is the point where the grph crosses the is, nd cn be found b letting = 0 (or f() = 0) nd solving for. The intercept of line is the point where the grph crosses the is, nd cn be found b letting = 0. Eponent Rules: If nd b re rel numbers nd m nd n re integers, then: roduct Rule: m n = m+n Zero Eponent: 0 = 1( 0) uotient Rule: m n = m n ( 0) Negtive Eponents: n = 1 ( 0) n ower Rules: ( m ) n = m n (b) m = m b m ( b ) m = m b m 4. TEST 4 MATERIAL olnomils A polnomil is finite sum of terms in which ll vribles hve epoents rised to nonnegtive integer powers nd no vribles pper in denomintor. A term is number or the product of number nd one or more vribles rised to powers. The numericl coefficient of term is the numericl fctor of the term.
3 3 The degree of term is the sum of the eponents on the vribles contined in the term. The degree of polnomil is the lrgest degree of ll its terms. Generl Shpes of Grphs of olnomil Functions Degree 2 Coefficient of 2 is positive number. Coefficient of 2 is negtive number. Degree 3 Degree 3 Coefficient of 3 is positive number Coefficient of 3 is negtive number Specil roducts erfect squre trinomil: ( + b) 2 = 2 + 2b + b 2 ( b) 2 = 2 2b + b 2 Difference of two squres: ( + b)( b) = 2 b 2 Sum nd difference of two cubes: ( + b)( 2 b + b 2 ) = 3 + b 3 ( b)( 2 + b + b 2 ) = 3 b 3 Finding the Gretest Common Fctor (GCF) of olnomil Step 1: Find the GCF of the numericl coefficients. Step 2: Find the GCF of the vrible fctors. Step 3: The product of the fctors found in Steps 1 nd 2 is the GCF of the monomils. Fctor b Grouping Step 1: Fctor out the GCF. Step 2: Group the terms so tht ech group hs common fctor. Step 3: Fctor out these common fctors. Step 4: Then see if the new groups hve common fctor. Fctor 2 + b + c b tril nd check Step 1: Fctor out the GCF. Step 2: Write ll pirs of fctors of 2 Step 3: Write ll pirs of fctors of c, the constnt term. Step 4: Tr combintions of these fctors until the middle term b is found. Step 5: If no combintion eists, the polnomil is prime.
4 4 5. TEST 5 MATERIAL Difference of two squres: 2 b 2 = ( + b)( b) erfect squre trinomil: ( + b) 2 = 2 + 2b + b 2 ( b) 2 = 2 2b + b 2 Zero Fctor ropert If nd b re rel numbers nd b = 0, then = 0 or b = 0. This propert is true for three or more fctors lso. Solve polnomil equtions b fctoring Step 1: Write the eqution so tht one side is 0. Step 2: Fctor the polnomil completel. Step 3: Set ech fctor equl to 0 using the zero fctor propert. Step 4: Solve the resulting equtions. thgoren Theorem In right tringle, the sum of the squres of the lengths of the two legs ( nd b) is equl to the squre of the length of the hpotenuse. (leg) 2 + (leg) 2 = (hpotenuse) 2 or 2 + b 2 = c 2 A c b B C Simplifing or Writing Rtionl Epression in Lowest Terms Step 1: Completel fctor the numertor nd denomintor of the rtionl epression. Step 2: Divide out fctors common to the numertor nd denomintor. (This is the sme s removing fctor of 1.) Multipling Rtionl Epressions The rule for multipling rtionl epressions is R S = R s long s 0 nd S 0 S To multipl rtionl epressions, ou m use the following steps: Step 1: Completel fctor ech numertor nd denomintor. Step 2: Use the previous rule nd multipl the numertors nd denomintors. Step 3: Simplif the product b dividing the numertor nd denomintor b their common fctors. Dividing Rtionl Epressions The rule for multipling rtionl epressions is R S = S R = S s long s 0, S 0, nd R 0. R To divide b rtionl epressions, use the rule bove to chnge division to multipliction b the reciprocl. Then simplif if possible. Adding or Subtrcting Rtionl Epressions with Common Denomintors If nd R re rtionl epressions, then + R = + R nd R = R Finding the Lest Common Denomintor (LCD) Step 1: Fctor ech denomintor completel Step 2: The LCD is the product of ll unique fctors ech rised to power equl to the gretest number of times tht the fctor ppers in n fctored denomintor. Add or subtrct rtionl epressions with unlike denomintors Step 1: Find the LCD of the rtionl epressions. Step 2: Write ech rtionl epression s n equivlent rtionl epression whose denomintor is the LCD found in Step 1. Step 3: Add or subtrct numertors, nd write the result over the common denomintor. Step 4: Simplif the resulting rtionl epression.
5 5 Rdicls s eponents: 1/n = n if n is rel number. 6. TEST 6 MATERIAL Frctions in eponents m/n = n m = ( n ) m m/n = 1 m/n = 1 n m = 1 ( n ) m Rules for Rdicls: (Sme s for eponents, s we cn write rdicls s eponents) If n nd n b re rel numbers roduct Rule: n n b = n b n ( ) uotient Rule: n = n ( 0) b b Distnce Formul: The distnce d between points ( 1, 1 ) nd ( 2, 2 ) is the following: d = ( 2 1 ) 2 + ( 2 1 ) 2 Midpoint Formul: The midpoint of the line between points ( 1, 1 ) nd ( 2, 2 ) is the following: ( 1 + 2, )
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