Unit 2: Rational Expressions

Transcription

1 Rational Epressions Pure Math 0 Notes Unit : Rational Epressions -: Simplifing Rational Epressions Rational Epressions: - fractions with polnomials as numerator and / or denominator. To Simplif (Reduce) Rational Epressions: (Factor all polnomials and reduce) Eample : Simplif the followings. a. ( ) ( ) ( ) ( ) 0 b. ( ) ( ) ( ) and c. 8 8 d. 8 8 ( ) ( )( ) 8 ( ) 0 ( ) ( )( ) ( ) 0 ( )( ) ( ) ( )( ) ( ) ( ) and ( ) ( ) ( ) 0 0 Page 0. Coprighted b Gabriel Tang, B.Ed., B.Sc.

2 Pure Math 0 Notes Rational Epressions e. 7 f. ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) and Cannot Simplif! ( ) 0 ( ) ( ) g. ( ) ( ) ( ) ( ) Multipl Both Sides b 0 ( ) ( ) - Homework Assignments Regular: pg. 8- # to (odd), 0, (a to e), AP: pg. 8- # to 8 (even), 0 to Coprighted b Gabriel Tang, B.Ed., B.Sc. Page.

3 Rational Epressions Pure Math 0 Notes Page. Coprighted b Gabriel Tang, B.Ed., B.Sc. -: Multipling and Dividing Rational Epressions Multipling Rational Epressions Eample : Simplif 7 Dividing Rational Epressions Eample : Simplif 8 8 ( ) ( ) 0 ( ) ( ) ,,, and 7 Reduce then Multipl! Reciprocal, then Reduce and Multipl! ( ) ( ) 0 ( ) ( ) 0 ( ) ( ) ,,, and NPV is taken from the numerator and denominator of the fraction after the sign

4 Pure Math 0 Notes Rational Epressions Eample : Simplif 7 Eample : Simplif ( )( ) ( ) 7 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) NPV is taken from the numerator and denominator of the fraction after the sign ( ) 0 ( ) ( ) ,, and - Homework Assignments Regular: pg. - # to 7 (odd), to, (a, b, c),, 8 AP: pg. - # to 8 (even), to, (a, b, c), to Coprighted b Gabriel Tang, B.Ed., B.Sc. Page.

5 Rational Epressions Pure Math 0 Notes -: Adding and Subtracting Rational Epressions (Part ) To Add and Subtract Rational Epressions. Find Common Denominator.. Obtain Equivalent Fractions.. Add or Subtract Numerators.. Reduce if Possible. ( ) ( ) 0 LCM and Equivalent Fractions Eample : Simplif the followings. a. b. LCM 7 7 ( ) Subtract the entire fraction. We will need brackets! LCM 0 7 ( ) 0 7 c. ( ) ( ) ( 7) LCM d. ( ) ( ) ( ) LCM 0 8 Page. Coprighted b Gabriel Tang, B.Ed., B.Sc.

6 Pure Math 0 Notes Rational Epressions Eample : A rectangle has a dimensions and. Find its perimeter. Perimeter l w Width P Length P ( ) ( ) ( ) P P P - Homework Assignments Regular: pg. 7 # to (odd),, 7, a AP: pg. 7 # to (even), to 8, a, 0 Coprighted b Gabriel Tang, B.Ed., B.Sc. Page.

7 Rational Epressions Pure Math 0 Notes -: Adding and Subtracting Rational Epressions (Part ) LCM of Monomial: - LCM of monomial coefficient, and the variable(s) with its / their highest eponent(s). Eample : LCM of a, a, a LCM of,, 0 Variable with Highest Eponent a LCM 0a Eample : LCM of,, 7 LCM of,, 7 Variables with Highest Eponents LCM LCM of Polnomial: - common factor(s) (written once) along with an uncommon (leftover) factor(s). Eample : LCM of and Factors of ( ) ( ) and Factors of ( ) ( ) Common Factor Leftover Factors LCM ( ) ( ) ( ) Eample : Simplif the followings. a. b. a b ab a b ( ) ( ) ( ) LCM b ( ) ab( ) ( ) b a b ab a b LCM a b a b 0 a 0 b 0 a 0 and b 0 Page. Coprighted b Gabriel Tang, B.Ed., B.Sc.

8 Pure Math 0 Notes Rational Epressions Coprighted b Gabriel Tang, B.Ed., B.Sc. Page 7. c. d. ( ) LCM ( ) () ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 ( ) ( ) ( ) 0 e. ( ) ( ) ( ) ( ) ( ) LCM ( ) ( ) ( ) Common Factor Leftovers ( ) 0 0 ( ) 0 ( ) 0 ± LCM LCM Switch Signs! ( )( ) 0 ( ) 0 ( ) 0

9 Rational Epressions Pure Math 0 Notes Eample : Car A takes hours to travel 00 km while car B takes hour less than car A to travel 00 km. a. Write an algebraic epression for each car A and car B that represent the speed at which the travel in terms of. b. Write an algebraic epression for their difference in speed. For questions involving Distance, Speed and Time, make a table to organize an data & epressions. a. Recall that: speed distance time Distance (km) Speed (km/hr) Time (hr) Car A 00 Car B b. Difference in Speed (Car B is faster) Speed of Car B Speed of Car A ( ) ( ) LCM ( ) ( ) 0 0 ( ) 0 In realit,, the time it takes for Car A has to be greater than ( ) ( ) km/hr - Homework Assignments Regular: pg. 7-7 # to (odd), to 7 AP: pg. 7-7 # to (even), to 7 Page 8. Coprighted b Gabriel Tang, B.Ed., B.Sc.

10 Pure Math 0 Notes Rational Epressions -: Review of Solving Linear Equations When Solving Linear Equations:. Epand and Simplif each side.. Make Sure ou LINE UP the Equal signs as ou work downward.. Collect Like-Terms with the Unknown terms on the Left Hand Side.. When Moving Terms to the other side of the Equal sign, do Reverse Order of Operation (SAMDEB) and Reverse Operations.. Solve for the solution b isolating the variable. Eample : Solve the following equations. a. 8 b. ( ) ( 7) 8 Collect Like Terms and 0 Reverse Operations Collect Like Terms and Reverse Operations c. ( ) ( ) Epand and Simplif each side Collect Like Terms and Reverse Operations - Homework Assignments Regular: pg # to 7 (odd), to 7 AP: pg # to 8 (even), to 8 Coprighted b Gabriel Tang, B.Ed., B.Sc. Page.

11 Rational Epressions Pure Math 0 Notes -7: Solving Rational Equations When solving rational equations, there are two methods ou can use. Method : Multipl Both Sides with LCM doing so will eliminate all denominators solve the remaining equation Method : Cross-Multiplication onl do so when there is a single fraction equals to another single fraction. solve the remaining equation. Eample : Solve the following equations. a. Method : Multipl Both Sides with LCM Method : Cross-Multiplication 0 8 ( ) 8 0 Multipl Both Sides b the LCM ( )( ) ( )( ) 8 8 Cross Multipl after Rearrangement Page 0. Coprighted b Gabriel Tang, B.Ed., B.Sc.

12 Pure Math 0 Notes Rational Epressions b. Method : Multipl Both Sides with LCM 8 Multipl Both Sides b the LCM Method : Cross-Multiplication Obtain LCM and Equivalent Fraction in the denominator can cancel on both sides Cross Multipl 8 ( ) 8 8 c NPV Method : Multipl Both Sides with LCM Method : Cross-Multiplication Multipl Both Sides b the LCM Cross Multipl ( )( ) ( )( ) ( ) ( ) Reduce when possible. ( ) ( ) NPV 0 0 Coprighted b Gabriel Tang, B.Ed., B.Sc. Page.

13 Rational Epressions Pure Math 0 Notes d. Method : Multipl Both Sides with LCM Method : Cross-Multiplication Multipl Both Sides b the LCM Obtain LCM Reduce and ( ) when Equivalent ( ) ( ) possible Fraction ( ) in the ( ) denominator can cancel on both sides ( ) ( ) NPV Homework Assignments Regular: pg. 8-8 # to (odd), 70 to 78 AP: pg. 8-8 # to 8 (even), 70 to 78 Page. Coprighted b Gabriel Tang, B.Ed., B.Sc.

14 Pure Math 0 Notes Rational Epressions -8: Problem Solving with Equations When Solving Word Problems:. Decide what the variable represents (usuall the unknown or the smaller item).. Set up an equation b reading the questions bit b bit, or organize the information on a table.. Solve and Verif.. Write out a final statement indicating the solution(s). Eample : Find three consecutive odd integers that have a sum of. Let smallest integer ( ) ( ) ( ) net odd integer ( ) largest odd integer 7 Therefore, the three integers are, 7 and 7 Eample : Mar has $7.8 in quarters and dimes. If she has 0 coins, how man coins of each tpe does she have? Let number of quarters 0 (0 ) 78 (0 ) number of dimes Each Quarter is worth cents Each Dime is worth 0 cents Mar has quarters and (0 ) dimes. Mar has quarters and dimes. ( $0. $0.0 $7.8) 8 8 Coprighted b Gabriel Tang, B.Ed., B.Sc. Page.

15 Rational Epressions Pure Math 0 Notes Eample : John went for a km mountain biking trip for two das. On da, he biked km/h faster than da. If he biked for hours on da while on da he biked for hours, how fast was he travelling on each da? With questions involving speed, distance, and time, we have to set up a table! Recall that: distance speed or time speed distance time Distance (km) Speed (km/hr) Time (hr) Da ( ) Da TOTAL ( ) John would have biked. km/h on da, and 8. km/h on da. -8 Homework Assignments Regular: pg. 0 - # to (odd), to 8, 0 to 0 Page. Coprighted b Gabriel Tang, B.Ed., B.Sc.

16 Pure Math 0 Notes Rational Epressions (AP) Eample : Mar and Jane each left Calgar and Edmonton respectivel at the same time, and drove towards Red Deer 0 km awa. If Mar drove 0 km/h faster than Jane and she had to wait 0 minutes before Jane arrived at Red Deer, how fast were both of them driving? Distance (km) Speed (km/hr) Time (hr) Mar 0 0 Jane (Faster Less Time) (Slower More Time) DIFFERENCE 0 minutes hr ( 0) 0 ( 0) ( 0) 00 ( 0) ( )( 00) ( 0) ( 0)( 00) 0 To Solve Quadratic Equations, bring everthing to one side and Factor! Speed cannot be Negative! 00 0 or Jane drove at 0 km/h, while Mar drove at 00 km/h -8 Homework Assignments AP: pg. 0 - # to 0 (even), to Coprighted b Gabriel Tang, B.Ed., B.Sc. Page.

17 Rational Epressions Pure Math 0 Notes -: Equations with Literal Coefficients Literal Coefficient: - the variable part of a monomial. Formula: - an equation where the variables represent certain measurements or constant. Eample : For d st, solve for s and t. d st d st t t s s d d s t t s When manipulating (rearranging) formulas, follow the rule of Reverse Order of Operation (SAMDEB) and Reverse Operations. Eample : Given the formulas below, solve for the variables indicated. a. lwh V Solve for H b. A πr Solve for r V lwh A r π V lw H A π r A r π c. V r π Solve for r d. SA πr (r H) Solve for H V r π SA r H πr V π r SA r H πr V r π Page. Coprighted b Gabriel Tang, B.Ed., B.Sc.

18 Pure Math 0 Notes Rational Epressions, where C is temperature in degree Celsius and F is temperature in Fahrenheit, to find the equivalent of o C in Fahrenheit. Eample : Use the formula, C ( F ) First, we have to manipulate (rearrange) the formula to solve for F. C F ( ) C F C F For o C, C and F? ( ) F F 77 - Homework Assignments Regular: pg. # to AP: pg. # to Coprighted b Gabriel Tang, B.Ed., B.Sc. Page 7.