Chapter 10. Rational Numbers

Transcription

1 Chapter 0 Rational Numbers The Histor of Chess 0. Rational Epressions 0. Multipling Rational Epressions 0.3 Dividing Rational Epressions 0. Dividing Polnomials 0.5 Addition and Subtraction of Rational Epressions with Like Denominators 0.6 Addition and Subtraction of Rational Epressions with Different Denominators 0.7 Rational Equations 0.8 Comple Fractions Chapter Review Chapter Test 63

2 Section 0. Rational Epressions Rational epressions are formed when we put polnomials in a fraction. In other words, rational epressions represent division. Simplif Because,, 0 and 0 are tied up in the trinomials through addition and subtraction, on first view it seems impossible that anthing could be reduced, or simplified. However, the idea in rational epressions is to find was to reduce the fraction b factoring. The above rational epression can be factored to: The move is obvious. Now can be cancelled and the rational epression becomes Simplif Simplif 5 5 Because the epressions are not eactl the same, the cannot be cancelled. However, 5 is the opposite of 5 and it ma be rewritten b factoring. Therefore, (5 ) turns into ( 5 + ) or ( 5) 5 And the new ratio is and can be reduced to Rational Epressions 65

3 Simplif Simplif 7 c 8c 7 Factor from the numerator c 8c 7 c 7 8 c 7 8 Practice: CONTENTS OF THIS PAGE HAVE BEEN REMOVED TO PROTECT COPYRIGHT 66 Chapter 0: Rational Numbers

4 Section 0. Multipling Rational Epressions Multiplication of rational epressions, like multiplication of fractions, is done b multipling numerator with numerator and denominator with denominator, and reduction to lowest terms. Multipl combine using one fraction and reduce Multipl 3z z - 9 z z Factor from (z ) reduce 3, 9, z, and (z ) a b Multipl a 5a 3 - a + a + b 3z z - 9 z z 3 z - 3z Factor and reduce: Numerator left: Factor difference of two squares. Numerator right: Factor trinomial (subtraction of two products). Denominators cannot be factored. reduce - a+ b a b a + a 3 multipl a + a+ b a b a 3 a 3a ab + 3b Practice: CONTENTS OF THIS PAGE HAVE BEEN REMOVED TO PROTECT COPYRIGHT 0. Multipling Rational Epressions 67

5 Section 0.3 Dividing Rational Epressions Division of rational epressions, similar to the division of fractions, is done b multipling the first fraction b the reciprocal of the second fraction. Once numerators and denominators are multiplied, reduce to lowest terms. Divide 3a 6 6a Flip (find reciprocal of) second fraction, reduce, and multipl what s left. 3a a 3 a a a a a a a a 7a z 6 + z - Divide 3 6 Flip second fraction, factor, and reduce: 3+ z z 3+ z z 3 + z z Divide Flip second fraction, factor and reduce: ( was factored out of the right denominator, making the trinomial positive, and the answer negative) Practice: CONTENTS OF THIS PAGE HAVE BEEN REMOVED TO PROTECT COPYRIGHT 68 Chapter 0: Rational Numbers

6 Section 0. Dividing Polnomials DIVIDING BY A MONOMIAL Divide - Each term of the numerator is divided b Answer: Divide Answer: DIVIDING BY A BINOMIAL Division of a polnomial b a binomial is set up and computed similarl to regular division, ecept that here we use terms with base, coefficient, and eponent. In other words, each term will be divided, then multiplied and subtracted. Sometimes there will be a remainder, sometimes the last operation will leave none. Divide ( + 8 0) ( ) Divide the leading term of the trinomial b the leading term of the binomial: ) and place the result above the line like regular division. ) Like in regular division, multipl trinomial. and place it below the leading term of the ) Do the same for the second term of the binomial, 0. Dividing Polnomials 69

7 ) ( ) 0 ) ( ) ) ( ) ) ( ) 0 0 (0 0) 0 Now subtract. The first term of the trinomial cancels and the second term becomes 0. This is the end of the first ccle. To start the second ccle, like in regular division bring down 0 and place it net to 0. 0 Divide and place answer above net to and multipl b 0. Multipl the second term, 0, of the answer b both terms of the binomial ( ) and place under (0 0) and subtract. remainder Answer: + 0 with 0 remaining Divide ( 3 + ) ( + ) + ) 3 + To properl divide an epression with missing terms like the binomial 3 +, the middle terms and must be inserted. This is done b leaving space for the missing terms before beginning. + ) ) ( 3 + ) + ) ( 3 + ) ( ) End of first ccle End of second ccle 70 Chapter 0: Rational Numbers

8 + + ) ( 3 + ) ( ) + ( + ) Answer End of third ccle. No remainder Practice: CONTENTS OF THIS PAGE HAVE BEEN REMOVED TO PROTECT COPYRIGHT 0. Dividing Polnomials 7

9 Section 0.5 Addition and Subtraction of Rational Epressions with Like Denominators Adding and subtracting rational epressions is no different from regular addition or subtraction of fractions: There must be a common denominator. ADDING AND SUBTRACTING WITH LIKE DENOMINATORS When the denominators are the same, place all polnomials over the same denominator, then add or subtract while following the rules of addition and subtraction of polnomials. Add Because the denominators are identical, then Subtract Because this is a subtraction, the ENTIRE second numerator must be written in parenthesis and subtracted: Add or subtract Write all trinomials over the same denominator. Because the third trinomial is subtracted, notice the sign changes Combine like terms and FACTOR to reduce Chapter 0: Rational Numbers

10 Add and/or subtract Write all trinomials over the same denominator. Because the third trinomial is subtracted, notice the sign changes factoring Factoring did not help to reduce. Answer remains - + Practice: CONTENTS OF THIS PAGE HAVE BEEN REMOVED TO PROTECT COPYRIGHT 0.5 Addition and Subtraction of Rational Epressions with like Denominators 73

11 Section 0.6 Addition and Subtraction of Rational Epressions with Different Denominators Fractions with different denominators cannot be added nor subtracted. To do so we must change them. Add The denominators show is common, but 3 is not. Correct b multipling second fraction times Complete addition using altered rational epression 3 3 (one) Add The denominators are not the same, even if it is onl a sign difference. The common denominator is ( + )( ). Multipl both fractions b one using the common denominator () () Add the results () and () Chapter 0: Rational Numbers

12 Subtract Factoring the first denominator: Because the second denominator is part of the second denominator, the first denominator is the common denominator. Multipl the second fraction b to get common denominator Subtract this result from the first fraction write parenthesis for subtraction Subtract Factoring both denominators: ( + )( + ) ( + )( + 6) Because ( + ) is common to both, the common denominator is ( + )( + )( + 6) Multipl both fractions b to get the common denominator Subtract the results of the second fraction from the results of the first fraction: (Continues in net page) 0.6 Addition and Subtraction of Rational Epressions with Different Denominators 75

13 Practice: CONTENTS OF THIS PAGE HAVE BEEN REMOVED TO PROTECT COPYRIGHT 76 Chapter 0: Rational Numbers

14 Section 0.7 Rational Equations A rational equation is one that contains fractions. Because fractions are divisions, one simple wa to start the solution of rational equations is b the use of multiplication. This will make for a less complicated solution as the fractions are eliminated. Solve The denominators show 0 is the common denominator for the equation; therefore, multipling each term of the equation b 0 will eliminate the rational numbers (fractions) reducing each term: Solve Multipl ever term b the common denominator : This is a quadratic equation: Write equal to zero, factor, and solve. 5 5 solve for : Answer: 5, Solve The denominators are not the same, even if it is onl a sign difference. The common denominator is ( + )( ). Multipl both sides b ( + )( ) reduce to: distribute: solve for : Rational Equations 77

15 Solve The common denominator is ( + 5)( 5), which happens to be also the difference of squares 5. Multipl all three terms b ( + 5)( 5) and reduce: distribute: combine: 0 9 Solve for b completing the square: Practice: Solve a 3a + 5 a + 3 3a a a a a c - + c z - z z p 3 p 5-3 p + 0 p + z z z Chapter 0: Rational Numbers

16 Section 0.8 Comple Fractions This a comple fraction: Basicall, these are large fractions made from fractions. To solve them, approach one section at a time. THE NUMERATOR The numerator of the mother fraction is an addition of fractions that reads: where the common denominator is ( ) 8 + Multipl left-hand fraction to adjust for the common denominator Add numerator fractions: () THE DENOMINATOR These denominator fractions are performing subtraction Multipl right-hand fraction to adjust for the common denominator: Subtract denominator fractions: () Dividing numerator and denominator Divide mother fraction (results from numerator () over results of denominator (): multipl b the reciprocal simplif a a a Comple Fractions 79

17 Numerator first: a a a + a 5 6a - 5 Divide results of numerator b denominator: 6a 3a a 8 8a a 5a 6-5 Simplif 3-9 Numerator first: 3 3 Then denominator: 9 9 Divide numerator b denominator. (Notice factoring of difference of squares) Practice: CONTENTS OF THIS PAGE HAVE BEEN REMOVED TO PROTECT COPYRIGHT 80 Chapter 0: Rational Numbers