Unit 8 Rational Functions

Transcription

1 Uit 8 Ratioal Fuctios Algebraic Fractios: Simplifyig Algebraic Fractios: To simplify a algebraic fractio meas to reduce it to lowest terms. This is doe by dividig out the commo factors i the umerator ad the deomiator. {This meas that you eed to factor both the umerator ad deomiator first.} o Commo factors divide to oe o Opposite factors divide to egative oe Multiplyig Algebraic Fractios: Factor each umerator ad deomiator. The rule for multiplyig algebraic fractios is the same as the rule for multiplyig umerical fractios multiply the tops (umerators) AND multiply the bottoms (deomiators). You ca oly cacel top with bottom or bottom with top. There is NO cacelig bottom with bottom or top with top.

2 Dividig Algebraic Fractios: The rule for dividig algebraic fractios is the same as the rule for dividig umerical fractios Chage the divisio sig to multiplicatio, take the reciprocal of the secod fractio ONLY, ad the follow the steps for multiplyig algebraic fractios. Addig/Subtractig Algebraic Fractios: The basic rule for addig ad subtractig fractios is to get a commo deomiator first {the smallest umber that both deomiators ca divide ito without remaiders}. Oce you get a commo deomiator, remember to just add/subtract umerators {keep the commo deomiator the same}. You might eed to factor the deomiators to help get the LCD. Whe rewritig fractios so that they have commo deomiators, remember whatever is multiplied times the bottom must ALSO be multiplied times the top. {Multiplyig the top ad bottom by the same umber is multiplyig by the multiplicative idetity elemet (which is = 1) ad therefore does ot chage the value of the fractio.}

3 Remider: {x 1 ad 1 x are opposites, therefore they divide to -1} x x Udefied Fractios: A fractio is udefied whe the deomiator equals zero. Solvig Fractioal Equatios: Method 1: Whe a sigle fractio is equivalet to a sigle fractio Cross multiply ad solve. Remember to check for extraeous roots by substitutig each aswer ito every deomiator to check for values that create udefied fractios. Method 2: Whe a sum or differece of fractios are ivolved Fid the LCD for each fractio ivolved i the equatio Oce you fid the LCD, multiply each fractio by the LCD to clear the deomiator Solve the resultig equatio Remember to check for extraeous roots by substitutig each aswer ito every deomiator to check for values that create udefied fractios. Method 1: Cross Multiply Method 2: Fid the LCD **must check** x = 0, 4 **must check** x = 2, oly!

4 Word Problems Ivolvig Ratioal Equatios Problems Ivolvig People or Machies Doig Work Whe problems ivolve two people workig together o a job the their rates add ad they ca perform the job workig together i a shorter amout of time. If we let x = time it takes 1 perso to complete the task, the his work rate is. I other words, he ca complete the 1 job i x umber of hours. Example: Bill s garde hose ca fill the pool i 10 hours. His eighbor has a hose tha ca fill the pool i 15 hours. How log will it take to fill the pool usig both hoses? Equatio eeded i order to solve this problem: Problems Ivolvig Percetages Whe problems ivolve percetages, thik about settig up a proportio. Example: So far i your volleyball practice, you have put ito play 37 of the 44 serves you have attempted. Fid the umber of cosecutive serves you eed to put ito play i order to raise your serve percetage to 90%. Equatio eeded i order to solve this problem: Problems Ivolvig Movig Objects Whe objects are i motio, a variatio of the distace formula must be used. This formula ca be maipulated i order to chage it to a formula that will give a ratioal expressio for the time. This variatio is Distace divided by Rate equals Time. Example: Adam drives 15 mph faster tha David does. Adam ca drive 100 miles i the same amout of time that David drives 80 miles. Fid Adams drivig speed. Table used to help setup equatio: Adam 100 Distace Rate Time David 80 Equatio eeded i order to solve this problem:

5 Uit 9 Ratioal Expoets ad Radical Fuctios RATIONAL/IRRATIONAL NUMBERS Ratioal Numbers: ca be expressed as a fractio LOOK LIKE: itegers, fractios, mixed umbers, repeatig decimals, termiatig decimals Irratioal Numbers: caot be expressed as a fractio LOOK LIKE: decimals that ever ed ad also ever repeat Perfect Squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169 Notatio for radicals: To simplify radicals: For square roots use the bleacher or factor tree method. For every two commo factors, oe comes out For radicals other tha square roots use the factor tree method. Look at the idex to determie how to group factors. Alterate method for square roots oly (idex of 2) look for the largest perfect square that is a factor of the umber. Radicals that are simplified have: o fractios left uder the radical symbol. o perfect power factors i the radicad, k. o expoets i the radicad, k, greater tha the idex,. o radicals appearig i the deomiator of a fractioal aswer. To add or subtract radicals: combie like terms the umber uder the radical sig must be the same ad the idex must be the same {you might eed to simplify each radical first to get like terms} To combie like terms, add or subtract the coefficiets of the radicals. Keep the commo umber uder the radical ad its idex the same.

6 Properties of Radicals: a a ab a b a a b b Multiplyig or Divide Radicals: Multiply/Divide the radicads (umber uder radical) together Multiply/Divide the coefficiets of the radicals together Simplify if possible Remember: radicals must have the same idex to multiply usig this method Ratioalizig the deomiator: A fractio that cotais a radical i its deomiator ca be writte as a equivalet fractio with a ratioal deomiator (a deomiator without a radical). You must ever leave a radical i the deomiator of a fractio. Whe the deomiator is a moomial (oe term), multiply both the umerator ad the deomiator by whatever makes the deomiator a expressio that ca be simplified so that it o loger cotais a radical. Whe there is more tha oe term i the deomiator, you will eed to multiply the umerator ad deomiator by the deomiator's cojugate. The cojugate is the same expressio as the deomiator but with the opposite sig i the middle, separatig the terms. Simplify: Simplify:

7 A expoetial expressio is oe which cotais a expoet. Fractioal expoets: b a a x x b missig a meas a = 2 missig b meas b = 1 (1) (2) (3) To solve equatios with fractioal expoets: Isolate the variable with the fractioal expoet. To elimiate the fractioal expoet, raise each side to the reciprocal power {remember ot to chage the sig}. Remember: If the deomiator of the reciprocal power is a eve umber, you iclude the aswer i your fial solutio. Solve the resultig equatio. Check your aswer(s) to avoid extraeous roots.

8 Radical Equatios: Isolate the radical to oe side of the equal sig. If the radical is a square root, square each side of the equatio. {If the radical is ot a square root, raise each side to a power equal to the idex of the root.} Solve the resultig equatio. Check your aswer(s) to avoid extraeous roots. Square both sides. x = 7, oly