Matapos ang Pagbubungkag ng Damikay tayo ay tutungo sa Rational Expressions MATH DIVISION, IMSP, UPLB
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1 Matapos ang Pagbubungkag ng Damikay tayo ay tutungo sa Rational Epressions 1
2 RATIONAL EXPRESSIONS Upon completion, you should be able to Simplify rational epressions Perform addition, subtraction, multiplication, and division of rational epressions Simplify comple fractions
3 RATIONAL EXPRESSIONS A rational epression is the ratio of polynomials. Thus if a and b are polynomials, a rational epression is of the form a b, where b 0 a is called the numerator, while b is called the denominator. 3
4 RATIONAL EXPRESSIONS The following are eamples of rational epressions: The following are NOT rational epressions: 1 1 4
5 Simplifying Rational Epressions Use the rule of cancellation ac bc a provided c 0 b Eamples: Simplify the following y 3 1 y 3 y 3, y ,
6 Simplifying Rational Epressions Eamples: Simplify the following y 16 8y y y y , 3, y 4 A rational epression is said to be in lowest terms if the numerator and denominator have no common factor ecept 1 and 1 (i.e., the polynomials in the numerator and denominator are relatively prime). 6
7 Multiplying Rational Epressions Use the rule: a c b d ac bd In using the rule, see to it that you cancel common factors first before multiplying the numerators and the denominators. 7
8 Multiplying Rational Epressions Eample: Find these products y 7y y, , 1, 3 8
9 Multiplying Rational Epressions Eample: Find these products. 3. 6c 9 c 3c 10 c 5 1 4c 3c 3 c 5 c c 5 c 5 4 c 3 3 c3 c, c 5 4c 5c 3 9
10 Multiplying Rational Epressions Remember: Sometimes, you have to factor out a ( 1) to be able to cancel factors and simplify y y 3y 3 5y 1y 3y 4y y 3 4y 3 4y y 3y y y 3 4y 3y 3 4y y 1 y y y,34 y 0, 3y 0, y 0 10
11 Dividing Rational Epressions Use the rule: a b c d a b d c ad bc Remember: To divide two rational epressions, multiply to the first the reciprocal of the second epression. 11
12 Dividing Rational Epressions Eample: Find the quotients and simplify: z 7 z 36y 1y z 1y 36y 7 z z,, y, z 0, 7,,5 1
13 Dividing Rational Epressions Eample: Find the quotients and simplify: 3. a a b a b b a a b a ab b a a b a ab b b a a b a b a a b a b b a a b a b a b b a a a b,a b Mathematics Division, IMSP, UPLB 13
14 Adding and Subtracting Rational Epressions Similar rational epressions are those that have the same denominators. The following are similar rational epressions: 1 3, 1 1,,
15 Adding and Subtracting Rational Epressions Two or more rational epressions can be made similar by getting their least common denominator (LCD). For eample: Find the LCD of 1, , Remember: The LCD is the epression with the smallest power that can be divided by the denominators eactly. 15
16 Adding and Subtracting Rational Epressions Remember: To find the LCD, 1. Factor each denominator completely.. Get all unique factors of the denominators. 3. Get the highest power of each factor appearing in the denominators and multiply them. 16
17 Adding and Subtracting Rational Epressions To make two rational epressions similar, we multiply both numerator and denominator of each epression by the factor that will make the denominators equal to the LCD. Eample: Convert to similar rational epressions. 1,
18 Adding and Subtracting Rational Epressions Eample:Convert to similar rational epressions. 1.. b y y, y y 3, 3 b a 16 a 4 a 1 3.,
19 Adding and Subtracting Rational Epressions To add(or subtract) two or more rational epressions, convert them to similar rational epressions then add(subtract) the numerators. The denominator of the sum(difference) is the LCD. In symbols, or a c b c a c b a c ad bc b d bd 19
20 Adding and Subtracting Rational Epressions Eamples: Find the sum or difference, then simplify the result
21 Adding and Subtracting Rational Epressions Eamples: Find the sum or difference, then simplify the result. 3. y y y y y y y y y y Final answer is... 6y y y 1
22 Adding and Subtracting Rational Epressions Eamples: Find the sum or difference, then simplify the result. 4. a1 a a 18 9a 3a a 1 9a 3a a a 18 a 18 9a 3a MADUGO!
23 Adding and Subtracting Rational Epressions Eamples: Find the sum or difference, then simplify the result. a1 a a 18 9a 3a a1 a a a3 a a1 a a a a a a 3a 3 3aa 3 a1 a Final answer is a 13a1 6a a 3 a 3 3
24 Adding and Subtracting Rational Epressions Eamples: Find the sum or difference, then simplify the result. 5. 4t t s s t s t t s t s 4t s t s t s t s t s t s t Final answer is... 4t s t, s t 4
25 Comple Fractions A comple fraction is the ratio of two or more rational epressions. To simplify a comple fraction, locate the main division bar and treat the problem as a division problem. Eample: Simplify y 1 1 y y, 0 y y 0 5
26 Comple Fractions , 0, 1 6
27 Comple Fractions , 0,1 Mathematics Division, IMSP, UPLB 7
28 WARNING! Ito ay mga malalaking pagkakamali wag gagayahin Mathematics Division, IMSP, UPLB
29 Summary In this section, we learned How to simplify a rational epression ac bc How to multiply and divide rational epressions a c ac b d bd a c a d ad b d b c bc How to add(subtract) rational epressions a c b c a b c a c ad bc b d bd a b 9
30 Summary That we should not divide by zero That factoring a 1 is sometimes needed to simplify a rational epression That in simplifying comple fractions, we should identify the main division bar and consider the problem as a division problem. 30
31 Assignment 1. Simplify the following rational epressions, if possible: a c b d MATHEMATICS DIVISION, IMSP, UPLB 31
32 Assignment. Find the following products and simplify the result b 1b 9 y 5 z y a. c b 48b 5 z 3y 15 a 1 a b. d. a 7 1 a 1 MATHEMATICS DIVISION, IMSP, UPLB 3
33 Assignment. Find the following products and simplify the result. 3 3 a y a y e. a y a a y y h. f. g t 11 t 1 t 10t 1 t 3t 8 10t 3t 1 3 w y w 6 4y y w 4 w y y 3 y y MATHEMATICS DIVISION, IMSP, UPLB 33
34 Assignment 3. Find the following quotients and simplify the result y 5 3a 1a a. c. 8z 1z 4b 7b 16 4 y 10y 1 9y 7 b. d. 0 5 y 14y 49 y 7 MATHEMATICS DIVISION, IMSP, UPLB 34
35 Assignment 3. Find the following quotients and simplify the result. e. h. f. g. w 3w 10 w 4w 5 w w 3 w 4w 3 a a 35 4a 8 3 3a 15a 9a a ay b by c cy d dy a b MATHEMATICS DIVISION, IMSP, UPLB 35
36 Assignment 4. Find the following sum/difference and simplify the result. a a a 7a c b m 4m d. 3 m 3 m 3 MATHEMATICS DIVISION, IMSP, UPLB 36
37 Assignment 4. Find the following sum/difference and simplify the result. e. f. g. h y4 4 y 1 5a a a 3a 1 a a 3 4a 4a 3 3 y y 1 y y y MATHEMATICS DIVISION, IMSP, UPLB 37
38 Assignment 5. Simplify the following comple fractions: 5b a. 6 b b. 3 m 1 n m n MATHEMATICS DIVISION, IMSP, UPLB 38
39 Assignment 5. Simplify the following comple fractions: y c. d y y MATHEMATICS DIVISION, IMSP, UPLB 39
40 Assignment 5. Simplify the following comple fractions: u u v 1 w 3 e. f. w w 1 3 u 1 v 1 1 MATHEMATICS DIVISION, IMSP, UPLB 40
41 Properties of the Set of Rational Epressions with + and 1. The sum/product of two rational epressions is a rational epression.. Addition/Multiplication of two rational epressions is associative. 3. Addition/Multiplication of two rational epressions is commutative. 4. Multiplication is distributive over Addition of rational epressions.
42 Properties of the Set of Rational Epressions with + and P P P P 0 and 1 Q Q Q Q 5. 0 is the additive identity. 1 is the multiplicative identity.
43 Properties of the Set of Rational Epressions with + and P P P P Q Q Q 0 P 6. The additive inverse of is Q P Q.
44 Properties of the Set of Rational Epressions with + and P Q Q P 1, P, Q 0 7. The multiplicative inverse of P Q Q is. P
45 Properties of the Set of Rational Epressions with + and The set of rational epressions, together with + and is a field.
46 Reflections 1. Is knowledge of factoring and special products important in simplifying rational epressions? Why?. How do you multiply rational epressions? 3. How do you divide rational epressions? 4. How do you find the LCD of two or more rational epressions? 5. How do you simplify comple fractions? 6. Give uses of rational epressions in real life and the operations defined on them. MATHEMATICS DIVISION, IMSP, UPLB 46
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