simplifying_rational_expressions with some examples

Simplifying polynomials

1. Operations on Rational
Expressions
Digital Lesson

2. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2
Rational expressions are fractions in which the
numerator and denominator are polynomials and the
denominator does not equal zero.
Example: Simplify .
3
9
2


x
x
)3(x
3
)3)(3(



x
xx
)3(
)3)(3(



x
xx
, x – 3  0
, x  3

3. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3
1. Factor the numerator and denominator of each
fraction.
2. Multiply the numerators and denominators of
each fraction.
4. Write the answer in simplest form.
3. Divide by the common factors.
To multiply rational expressions:
d
c
b
a
•
bd
ac


4. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4
)3)(1(
)2(



xx
xx
Factor the numerator and
denominator of each fraction.
Divide by the common factors.
Write the answer in simplest
form.
)1)(3(
)3(



xx
xx
)1)(3(
)2)(1(

•


xx
xx
32
3
2
2


xx
xx
32
2
2
2

•


xx
xx
Multiply .

Example:
)1)(3)(1)(3(
)2)(1)(3(



xxxx
xxxx
)1)(3)(1)(3(
)2)(1)(3(



xxxx
xxxx
Multiply.

5. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5
b
a
a
b
1. Multiply the dividend by the reciprocal of the
divisor. The reciprocal of is .
2. Multiply the numerators. Then multiply the
denominators.
4. Write the answer in simplest form.
3. Divide by the common factors.
To divide rational expressions:
d
c
b
a

c
d
b
a
•
bc
ad


6. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6
Example: Divide .
2
22
22
z
yxx
z
yxx 


yxx
z
z
yxx
2
22
22
•



zxyx
zxyx
•
•
)1(2
)1(
2



zxyx
zxyx
•
•
)1(2
)1(
2



Multiply by the reciprocal
of the divisor.
Factor and multiply.
Divide by the common
factors.
Simplest form
2
z


7. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7
The least common multiple (LCM) of two or more numbers is the
least number that contains the prime factorization of each number.
Examples: 1. Find the LCM of 10 and 4.
2. Find the LCM of 4x
2
+ 4x and x
2
+ 2x + 1.
4x
2
+ 4x = (4x)(x +1) = 2 • 2 x (x + 1)
x
2
+ 2x + 1 = (x +1)(x +1)
LCM = 2 • 2 x (x +1)(x +1)
factors of 4x
2
+ 4x
factors of x
2
+ 2x + 1
10 = (5 • 2)
LCM = 2 • 2 • 5
factors of 10
factors of 4
4 = (2 • 2)
= 4x
3
+ 8x
2
+ 4x
= 20

8. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8
LCM
Fractions can be expressed in terms of the least common
multiple of their denominators.
Example: Write the fractions and in terms of the
LCM of the denominators.
xx
x
126
12
2


2
4x
x
)2(6
12



xx
x
)2)(2( xx
x

)2(3
)2(3
•


x
x
x
x
2
2
•
2
4x
x
xx
x
126
12
2


The LCM of the denominators is 12x
2
(x – 2).
)2(12
))(2(3
2



xx
xx
)2(12
)12(2
2



xx
xx

9. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9
1.If necessary, rewrite the fractions with a common
denominator.
To add rational expressions:
b
c
b
a

b
ca

To subtract rational expressions:
2. Add the numerators of each fraction.
1.If necessary, rewrite the fractions with a common
denominator.
2. Subtract the numerators of each fraction.
b
c
b
a

b
ca


10. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10
Example: Add .
14
5
14
2 xx

14
52xx

14
7x

2
x

Example: Subtract .
4
4
4
2
22


xx
x
4
42
2



x
x
)2)(2(
)2(2



xx
x
)2(
2


x

11. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11
Two rational expressions with different denominators can be
added or subtracted after they are rewritten with a common
denominator.
Example: Add .
4
6
2
3
22




xxx
x
)2)(2(
6

)2(
3





xxxx
x
)2(
)2(
•


x
x
)(
)(
•
x
x
)2)(2(
6)2)(3(



xxx
xxx
)2)(2(
66
2



xxx
xxx
)2)(2(
65
2



xxx
xx
)2)(2(
)1)(6(



xxx
xx
)2)(2(
6

)2(
3





xxxx
x

12. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12
Example: Subtract .
1
1
1
22
2


xx
x
1
1


2
2



x
x
)1 )(1(
)1)(1(



xx
xx
)1)(1(
)1)(1(



xx
xx
1
Add numerators.
Factor.
Divide.
Simplest form