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MATH DIVISION, IMSP, UPLB
Matapos ang
“Pagbubungkag ng
Damikay” tayo ay tutungo
sa
Rational Expressions
1
MATH DIVISION, IMSP, UPLB
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
MATH DIVISION, IMSP, UPLB
RATIONAL EXPRESSIONS
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
MATH DIVISION, IMSP, UPLB
RATIONAL EXPRESSIONS
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
MATH DIVISION, IMSP, UPLB
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
MATH DIVISION, IMSP, UPLB
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
MATH DIVISION, IMSP, UPLB
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
MATH DIVISION, IMSP, UPLB
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
Multiplying Rational Expressions
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
MATH DIVISION, IMSP, UPLB
Example: Find these products.
9
Multiplying Rational Expressions
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
MATH DIVISION, IMSP, UPLB
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
Multiplying Rational Expressions
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
MATH DIVISION, IMSP, UPLB
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
MATH DIVISION, IMSP, UPLB
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
Dividing Rational Expressions
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:
Dividing Rational Expressions
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
MATH DIVISION, IMSP, UPLB
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
MATH DIVISION, IMSP, UPLB
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
Adding and Subtracting Rational Expressions
1. 2.
MATH DIVISION, IMSP, UPLB
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
Adding and Subtracting Rational Expressions
MATH DIVISION, IMSP, UPLB
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
Adding and Subtracting Rational Expressions
20
7,
2
1
MATH DIVISION, IMSP, UPLB
Example:Convert to similar rational expressions.
2 2
2 2
x y x y,
x y x y
18
Adding and Subtracting Rational Expressions
1.
2.
162
3,
4
322
ab
a
ab
2 13. ,
3 3x x
MATH DIVISION, IMSP, UPLB
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
Adding and Subtracting Rational Expressions
c
ba
c
b
c
a
MATH DIVISION, IMSP, UPLB
Examples: Find the sum or difference, then simplify the result.
1 2
2 3
2
1 2
x x
20
Adding and Subtracting Rational Expressions
1.
2.
3 4 7
6 6
222
22
x
x
xx
x
MATH DIVISION, IMSP, UPLB
Examples: Find the sum or difference, then simplify the result.
2 2
2 2
x y x y
x y x y
21
Adding and Subtracting Rational Expressions
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
MATH DIVISION, IMSP, UPLB
Examples: Find the sum or difference, then simplify the result.
2 2
1 2
2 18 9 3
a a
a a a
22
Adding and Subtracting Rational Expressions
4.
MADUGO!
22
22
39182
1822391
aaa
aaaaa
MATH DIVISION, IMSP, UPLB
Examples: Find the sum or difference, then simplify the result.
2 2
1 2
2 18 9 3
a a
a a a
23
Adding and Subtracting Rational Expressions
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
MATH DIVISION, IMSP, UPLB
2
2 2
4t t s s t
t s t ss t
24
Adding and Subtracting Rational Expressions
Examples: Find the sum or difference, then simplify the result.
5.
24t s t s t
s t s t s t s t
Final answer is... 4t
s t
, s t
MATH DIVISION, IMSP, UPLB
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
MATH DIVISION, IMSP, UPLB
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
MATH DIVISION, IMSP, UPLB
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
MATH DIVISION, IMSP, UPLB
• 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
MATHEMATICS DIVISION, IMSP, UPLB 32
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
MATHEMATICS DIVISION, IMSP, UPLB 33
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
MATHEMATICS DIVISION, IMSP, UPLB 34
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
MATHEMATICS DIVISION, IMSP, UPLB 35
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
MATHEMATICS DIVISION, IMSP, UPLB 36
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
MATHEMATICS DIVISION, IMSP, UPLB 37
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
MATHEMATICS DIVISION, IMSP, UPLB 38
2
2
2
51
62
3
b m
na. b.b m
n
5. Simplify the following complex fractions:
Assignment
MATHEMATICS DIVISION, IMSP, UPLB 39
22
25
2
3 2 2
2 32 1
3 12 21
4
xx
yxc. d .x x x
x x yy
5. Simplify the following complex fractions:
Assignment
MATHEMATICS DIVISION, IMSP, UPLB 40
1 2
1 2
13 2
1 2 3
1
uu
v
x xwe. f .w x xw
u
v
5. Simplify the following complex fractions:
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
Properties of the Set of Rational
Expressions with + and ●
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
Properties of the Set of Rational
Expressions with + and ●
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
Properties of the Set of Rational
Expressions with + and ●
The set of rational expressions,
together with + and is a field.
Properties of the Set of Rational
Expressions with + and ●
MATHEMATICS DIVISION, IMSP, UPLB 46
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.