Matapos ang Pagbubungkag ng Damikay tayo ay tutungo sa ...jfrabajante.weebly.com/uploads/1/1/5/5/11551779/11_math_17.pdf · Matapos ang “Pagbubungkag ng Damikay” tayo ay tutungo

MATH DIVISION, IMSP, UPLB

Matapos ang

“Pagbubungkag ng

Damikay” tayo ay tutungo

sa

Rational Expressions

1


RATIONAL EXPRESSIONS

Upon completion, you should be able to

• Simplify rational expressions

• Perform addition, subtraction, multiplication, and division of rational expressions

• Simplify complex fractions

2



A rational expression is the ratio of polynomials.

Thus if a and b are polynomials, a rational

expression is of the form

a

b, where

a is called the numerator, while

3

b is called the denominator.

0b



The following are examples of rational expressions:

1

2 2

2

x

2

3

2 3

5

x x

x x

5

2

1

2

x

x x

The following are NOT rational expressions:

2

x

1

1

x

x

4


Simplifying Rational Expressions

Examples: Simplify the following.

2

2

6 17 7

12 13 35

x x

x x

Use the rule of cancellation

ac a

bc b provided c 0

5

2 5

3 2

7

21

x y

x y

1.

2.

3

3

y

x

3 7 2 1

3 7 4 5

x x

x x

2 1

4 5

x

x

, 0y

,3 7 0x


Examples: Simplify the following.

3

2

27

2 15

x

x x

2

2

16

8 2

y

y y

A rational expression is said to be in lowest terms if the numerator and denominator have no common factor except 1 and –1 (i.e., the polynomials in the numerator and denominator are relatively prime).

6

Simplifying Rational Expressions

3.

4.

23 3 9

3 2 5

x x x

x x

2 3 9

2 5

x x

x

4

2

y

y

, 3x

, 4y


Multiplying Rational Expressions

Use the rule:

In using the rule, see to it that you cancel common factors first before multiplying the numerators and the denominators.

a c ac

b d bd

7


Example: Find these products.

3 2

4 2

5 21

7 25

x y

y x

2

2

12 3 3

5 5 9

x x x

x x

8


1.

2.

2

3

5

x

y

4 3 3 1

5 1 3 3

x x x

x x x

3 4

5 3

x

x

, 0x

, 1, 3x x


Example: Find these products.

9


3. 2

2

6 9 3 10

12 425

c c c

cc

3 2 3 5 2

5 5 4 3

c c c

c c c

3 2 3 2

4 5 3

c c

c c

, 5c


2 2 2

2 2 2

9 16 2 3

23 5 12 3 4

x y xy x x y

x yx xy y xy y

Remember: Sometimes, you have to factor out a (–1) to be able to

cancel factors and simplify.

10


4.

3

2

3 4 3 4 2

3 4 3 3 4

x x y

y x y

x y x y y x

x y x y x y

1

2

2x

y x y

x y

x

y

,3 4 0,

3 0,

2 0

x y

x y

x y


Dividing Rational Expressions

Use the rule:

a c

b d

a d

b c

ad

bc

Remember: To divide two rational expressions, multiply to the

first the reciprocal of the second expression.

11


Example: Find the quotients and simplify:

3 2 281 27

36 12

xz x z

y xy

2 2

2 2

9 14 3 10

4 21 2 35

x x x x

x x x x

12


1.

2.

3

2 2

81 12

36 27

xz xy

y x z z

2 2

2 2

9 14 2 35

4 21 3 10

x x x x

x x x x

2 7 7 5

3 7 5 2

x x x x

x x x x

, , , 0x y z

, 7, 2,5x 3

7

x

x

Mathematics Division, IMSP, UPLB 13

2 2 2 2

2 22

a a b a b

b a a b a ab b

Example: Find the quotients and simplify:


3.

2 2 2 2

2 2

2a a b a ab b

b a a b a b

22 2 a ba a b

b a a b a b a b

2 2

2

a a b

a b

,a b


Adding and Subtracting Rational Expressions

Similar rational expressions are those that have the same denominators.

The following are similar rational expressions:

1 3

2 2,

1 1x,

x x

2 2

2

1 1

x x,

x x

14


Two or more rational expressions can be made similar by getting their least common denominator (LCD).

For example: Find the LCD of

1 2

2 3,

2

1 2,

x xRemember: The LCD is the expression with the smallest power that can be divided by the denominators exactly.

15


1. 2.


Remember: To find the LCD,

1. Factor each denominator completely.

2. Get all unique factors of the denominators.

3. Get the highest power of each factor appearing in the denominators and multiply them.

16



To make two rational expressions similar, we multiply both numerator and denominator of each expression by the factor that will make the denominators equal to the LCD.

Example: Convert to similar rational expressions.

17


20

7,

2

1


Example:Convert to similar rational expressions.

2 2

2 2

x y x y,

x y x y

18


1.

2.

162

3,

4

322

ab

a

ab

2 13. ,

3 3x x


To add(or subtract) two or more rational expressions, convert them to similar rational expressions then add(subtract) the numerators. The denominator of the sum(difference) is the LCD. In symbols,

or a c ad bc

b d bd

19


c

ba

c

b

c

a


Examples: Find the sum or difference, then simplify the result.

1 2

2 3

2

1 2

x x

20


1.

2.

3 4 7

6 6

222

22

x

x

xx

x



2 2

2 2

x y x y

x y x y

21


3.

2 2 2 2

2 2

x y x y x y x y

x y x y

Final answer is...

6

2 2

xy

x y x y



2 2

1 2

2 18 9 3

a a

a a a

22


4.

MADUGO!

22

22

39182

1822391

aaa

aaaaa



2 2

1 2

2 18 9 3

a a

a a a

23


4. 2

1 2

3 32 9

a a

a aa

1 2

2 3 3 3 3

a a

a a a a

1 2

2 3 3 3 3

a a

a a a a

Final answer is…

2 13 12

6 3 3

a a

a a a


2

2 2

4t t s s t

t s t ss t

24



5.

24t s t s t

s t s t s t s t

Final answer is... 4t

s t

, s t


Complex Fractions

A complex fraction is the ratio of two or more rational expressions. To simplify a complex fraction, locate the main division bar and treat the problem as a division problem.

1 1

1 1

x y

x y

Example: Simplify.

25

1. y x

y x

, 0

0

x

y


1 1

1 11 1

1 1

x x

x x

x x

26

Complex Fractions

2.

2 21 1

1 1

x x

x x

1 1 2x x , 0, 1x

Mathematics Division, IMSP, UPLB 27

11

11

11

x

3.

Complex Fractions

11

11

1x

x

11

11

x

x

11

1

1

x x

x

2 x , 0,1x

Mathematics Division, IMSP, UPLB

WARNING!

Ito ay mga malalaking

pagkakamali… ‘wag

gagayahin.

1011

1

1

1x

1

1

x

x

x

x

xx

x


Summary

In this section, we learned

• How to simplify a rational expression ac a

bc b

• How to multiply and divide rational expressions

a c ac

b d bd

a c a d ad

b d b c bc

• How to add(subtract) rational expressions

a c ad bc

b d bd

29

c

ba

c

b

c

a


• That we should not divide by zero

• That factoring a “–1” is sometimes needed to simplify a rational expression

• That in simplifying complex fractions, we should identify the main division bar and consider the problem as a division problem.

30

Summary

MATHEMATICS DIVISION, IMSP, UPLB 31

2

2

45 7

145 7

7 9 3

349

xa. c.

x

x xb. d .

xx

1. Simplify the following rational expressions, if possible:

Assignment


7 5 2 2 2

3 9 3

2 2 2

12 12 9 25

3 153 48 5

1 7 2 1 4 4

7 1 2 1

b b x y x z xya. c.

xyb b x z

a a x x x xb. d.

a a x x

2. Find the following products and simplify the result.

Assignment


3 3

2 2

2

2

2 2

2 2

2

2

5 24 2 2

8 86

2 11 12 10 21

3 28 10 32 12

3 2 6 4 2

2 4 2 3 2

a y a ye.

a y a a y y

x x xh.

xx x

t t t tf .

t t t t

x w y w x y x x yg.

x w x w y y x y y

2. Find the following products and simplify the result.

Assignment


2 4 5

2

2 2

2 2 2

7 5 3 12

8 12 42 7

16 4 10 21 9 27

520 14 49 7

x y x a aa. c.

z z b b

x x y y yb. d.

xx x y y y

3. Find the following quotients and simplify the result.

Assignment


2 2

2 2

2

2 3

2 2 2

3 10 4 5

2 3 4 3

2 35 4 28

3 15 9

2

3 12 9 14 4 4

5 10 3 21 3 1

w w w we.

w w w w

a a ah.

a a a

ax ay bx by cx cy dx dyf .

a b

x x x x xg.

x x x

3. Find the following quotients and simplify the result.

Assignment


8 5 3 1 7

9 9 5 10 25

1 1 1 2 42

2 3 3 3

x x a a aa. c.

x x

m mb. d.

x x m m

4. Find the following sum/difference and simplify the result.

Assignment


2 2 2

2 2 2

4 5 1

1 2 1

3 2

4 4

1 5 2

2 3 1 2 3 4 4 3

3 2 1

x xe.

x x x

f .y y

a ag.

a a a a a a

x y yh.

x yx y x xy

4. Find the following sum/difference and simplify the result.

Assignment


2

2

2

51

62

3

b m

na. b.b m

n

5. Simplify the following complex fractions:

Assignment


22

25

2

3 2 2

2 32 1

3 12 21

4

xx

yxc. d .x x x

x x yy


Assignment


1 2

1 2

13 2

1 2 3

1

uu

v

x xwe. f .w x xw

u

v


Assignment

Properties of the Set of Rational

Expressions with + and ●

1. The sum/product of two rational

expressions is a rational expression.

2. Addition/Multiplication of two rational

expressions is associative.

3. Addition/Multiplication of two rational

expressions is commutative.

4. Multiplication is distributive over

Addition of rational expressions.

0 and 1

5. 0 is the additive identity.

1 is the multiplicative identity.

P x P x P x P x

Q x Q x Q x Q x



0

6. The additive inverse of is

.

P x P xP x P x

Q x Q x Q x

P x

Q x

P x

Q x



1, , 0

7. The multiplicative inverse of

is .

P x Q xP x Q x

Q x P x

P x

Q x

Q x

P x



The set of rational expressions,

together with + and is a field.




Reflections

1. Is knowledge of factoring and special products important in simplifying rational expressions? Why?

2. How do you multiply rational expressions? 3. How do you divide rational expressions? 4. How do you find the LCD of two or more rational

expressions? 5. How do you simplify complex fractions? 6. Give uses of rational expressions in real life and the

operations defined on them.

Documents

Matapos ang Pagbubungkag ng Damikay tayo ay tutungo sa ...jfrabajante.weebly.com/uploads/1/1/5/5/11551779/11_math_17.pdf · Matapos ang “Pagbubungkag ng Damikay” tayo ay tutungo