A.1 Algebra Review: Polynomials/Rationals. Definitions:

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1 MATH 040 Notes: Uit 0 Page 1 A.1 Algera Review: Polyomials/Ratioals Defiitios: A polyomial is a sum of polyomial terms. Polyomial terms are epressios formed y products of costats ad variales with whole umer epoets. Moomials, iomials, ad triomials are oe-term, two-term, ad three-term polyomials, respectively. For the rest of this page, assume that all terms refereced are polyomial terms. The degree of a term is equal to the sum of the epoets i the term. The degree of a polyomial is equal to the maimum value of all the degrees of a term i the polyomial. For a polyomial of oe variale, the leadig term is the term of the highest degree. A coefficiet is the costat factor of a term (as opposed to the variale factors). The leadig coefficiet is the coefficiet of the leadig term. A costat term is a term without a eplicit variale factor. A ratioal epressio is a quotiet of polyomials. E 1: Aswer the followig questios aout the polyomial. 4 y3y 5y For parts a ad c, assume that the idetified term is ased o the order from left to right. a. What is the degree of the secod term writte?. What is the degree of the polyomial? c. What is the coefficiet of the fourth term writte? E : Aswer the followig questios aout the polyomial fuctio f a. What is the degree of f?. What is the leadig term of f? c. What is the leadig coefficiet of f? d. What is the costat term of f?

2 MATH 040 Notes: Uit 0 Page Shortcuts for epadig special iomial products: a a a a a a a a a E 4: Epad. a E 3: Epad y c. Recall that factorig is divisio without elimiatig the divisor. A polyomial that is t factorale is said to e prime. E 5: Factor. 16 Whe factorig, always factor a greatest commo factor (GCF) first, if oe eists. To factor four term polyomials, group pairs of terms ad fid the GCF of each group; the resultig pairs of products may e factorale agai. E 6: Factor To factor triomials of the form ² ± ± c, rememer to fid the factors of c which either add or sutract (ased o the sig of c) to make.

3 MATH 040 Notes: Uit 0 Page 3 E 7: Factor. a.. E 8: Factor. a A ratioal epressio ca e simplified y cacelig commo factors of the umerator ad deomiator. The resultig epressio is equivalet i value everywhere ecept for the value that made the commo factor zero. E 9: Simplify Equatios ca e solved y factorig usig the Zero Product Property (If A B = 0, the A = 0 or B = 0). E 11: Solve E 10: Solve

4 MATH 040 Notes: Uit 0 Page 4 A. Algera Review: Lies Defiitio: The slope of a lie is a ratio of vertical chage to horizotal chage over ay segmet of the lie. The value of the slope descries the directio ad the steepess of the lie. The directio of a lie ca e idicated y the eistece ad sig of the slope: Positive slopes idicate the lie is icreasig (goig up) from left to right. Negative slopes idicate the lie is decreasig (goig dow) from left to right. A lie with a slope of zero is horizotal. A lie without slope is vertical. The slope of a lie, epressed as m, ca e calculated y kowig two poits ( 1, y 1 ) ad (, y ) y the formula: y y m 1 1 It also ca e foud visually y usig two poits of referece o the lie ad measurig the vertical ad horizotal chage as descried y the slope defiitio. E 1: Fid the slope of a lie passig through E 3: Fid the slope of each lie o the graph. 1 E : Fid the slope of a lie passig through

5 MATH 040 Notes: Uit 0 Page 5 All lies ca e epressed i the geeral form of A + By = C, where A, B, ad C are real umers with A ad B ot simultaeously zero. Lies are more commoly epressed i other forms which ifer their ehavior. Vertical lies are lies of the form = a, where a is ay real umer. Horizotal lies are lies of the form y =, where is ay real umer. Olique lies are lies of the form y = m +, where m is the o-zero slope of the lie ad is ay real umer. The form y = m + is ofte called the slopeitercept form ecause, i additio to m represetig the slope, idicates the y-value where the lie crosses the y-ais (y-itercept). Aother useful form of lies is y y 1 = m( 1 ) which is called poit-slope form ecause, i additio to m represetig the slope, ( 1, y 1 ) idicates a poit the lie passes through. A useful variatio o this form ivolves solvig for y: y = m( 1 ) + y 1. The slope-itercept form of a lie is uique while the poit-slope form is ot (ut ca e useful y the iformatio it commuicates). E 4: Fid the equatio of a horizotal lie passig through (, 8). E 7: Fid the slope-itercept form of a lie E 5: Fid the equatio of a lie which passes E 6: Fid the slope-itercept form of a lie

6 MATH 040 Notes: Uit 0 Page 6 E 8: Fid the slope-itercept form of lie 1 ad a poit-slope form of lie o the graph. E 9: Sketch a graph of the equatio y = 1. 1

7 MATH 040 Notes: Uit 0 Page 7 A.3 Algera Review: Epoets/Epoetials Further laws of epoets (where Laws of epoets (ote coditios): 0 1 Power of Zero m m Product Rule m m Quotiet Rule m m Power Rule a a Distriutive over Product a a, 0 Distriutive over Quotiet m m 1 Negative Epoets m m m Ratioal Epoets Coditios: 1: Bases & a are ozero real ad epoets m & are itegers : Bases & a are positive real ad epoets m ad are real. E 1: Rewrite each epressio without usig a quotiet. 4 a. 3 E : Epress each epressio with a positive epoet ad evaluate if = 4. a c. 5 y c. 3

8 MATH 040 Notes: Uit 0 Page 8 E 3: Epress each radical epressio with a ratioal epoet. E 4: Epress each epressio with a radical ad evaluate if = 64. a. 3 4 a c d. 7 5 c. 1 6 A power term is a epressio ivolvig epoets where the ase is a variale ad the epoet is fied. Eamples would iclude ,, ad. Polyomial terms are power terms whose epoet is a atural umer ad are multiplied y a coefficiet. A epoetial term is a epressio ivolvig epoets where the ase is fied ad the epoet is a variale. Eamples would iclude ad 10. A special epoetial term is the atural epoetial term, e, a epoetial term whose ase is the umer e. The umer e, referred to as the atural ase, is a irratioal umer with a approimate value of.718. e is defied y the result of the epoet gettig ifiitely large i the 1 epressio 1. The iverse of a power term is aother power 1 3 term (for istace, the iverse of is 3).

9 MATH 040 Notes: Uit 0 Page 9 I algera we are iterested i iverses ecause they allow you to go reverse a process. The iverse of a epoetial term is a logarithmic term. A logarithmic term equals a epoet formed y the relatioship of a ase ad a iput. Eamples iclude log ad log 10, the latter called a commo logarithm ad ofte epressed without the ase. Solvig a equatio ivolvig a atural epoetial requires a atural logarithm to e used. Occasioally factorig will also e importat to apply the Zero Product Property. E 5: Solve. Preset the solutio i eact form ad approimated to five decimal places. 4e 1 A special logarithmic term is the atural logarithmic term, l, a logarithm whose ase is the umer e. Recall that l 1 = 0 ad l e =. E 6: Solve. Preset the solutio i eact form ad approimated to five decimal places. 3 3e 0 E 7: Solve. Preset the solutio i eact form ad approimated to five decimal places. 4l 1

10 MATH 040 Notes: Uit 0 Page 10 A.4 Algera Review: Fuctios Defiitios: A fuctio is a relatio defied y a rule where each iput creates at most oe output. The domai of a fuctio is the collectio of values that ca e iputs of the fuctio. The atural domai of a fuctio is the largest domai possile without coditios applied. The rage of a fuctio is the collectio of values that ca e outputs of the fuctio. Polyomial ad epoetial fuctios have atural domais of all real umers. Radical fuctios f ( ) z have a domai of real umers such that z > 0. 1 Ratioal fuctios f ( ) have a domai of z real umers ecludig solutios to z = 0. Logarithmic fuctios f ( ) log a z have a domai of real umers such that z > 0. A reveue fuctio R() calculates the total reveue of sellig uits of a item. Some reveue fuctios are epressed as the product of a price fuctio p() ad a quatity fuctio q(). That is, R = p q. A demad fuctio ca e ased o price or quatity. For istace, if price is the result of the quatity demaded, the demad fuctio ca e epressed as p = D(q). I some cases through demad is structured where quatity sold is the result of a give price: q = D(p). A cost fuctio C() determies the total cost of producig uits of a item. A fied cost is a costat quatity of a cost fuctio; i other words C(0) equals the fied cost. Margial cost is the amout it costs to produce oe more uit. I a liear fuctio the margial cost equals the slope. Ay category of fuctio ca have a aalogous margial. A profit fuctio P() determies the total profit of producig ad sellig uits of a item. P = R C. The reak-eve poit is the poit where profit is zero. That is, P() = 0.

11 E 1: Based o past eperiece a compay kow that chargig $600 for a semiar o maagemet techiques will attract 1000 people. For each $0 decrease i the fee a additioal 100 people will atted the semiar. Fid a price fuctio ad a quatity fuctio ased o, the umer of times they decrease the semiar fee y $0. The fid the reveue fuctio i terms of. MATH 040 Notes: Uit 0 Page 11 E : The same compay as i eample 1 has foud that the cost of puttig o the semiar is $100 per perso attedig plus additioal geeral overhead charges of $ Fid the cost fuctio ad profit fuctio ased o. The fid ad iterpret the profits eared y pricig the semiar at $500.

12 MATH 040 Notes: Uit 0 Page 1 E 3: Fid the liear cost fuctio if the fied cost is $400 ad the total cost of producig 60 items is $00. E 4: For uits produced, cosider reveue ad cost fuctios of R ( ) 4 ad C ( ) 3 0, respectively, each i dollars. Fid the average profit fuctio ad the fid ad iterpret whe 5 uits are produced ad sold. A average reveue fuctio R, calculated y R/, determies the average reveue geerated for each item sold. There are aalogous average cost fuctios P C ad average profit fuctios