Transcript Operations on Rational Expressions

Digital Lesson
Operations on Rational
Expressions
Rational expressions are fractions in which the
numerator and denominator are polynomials and the
denominator does not equal zero.
2
x

9
Example: Simplify
.
x 3
( x  3)( x  3)

x 3
( x  3)( x  3) , x – 3  0

( x  3)
 ( x  3), x  3
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To multiply rational expressions:
1. Factor the numerator and denominator of each
fraction.
2. Multiply the numerators and denominators of
each fraction.
3. Divide by the common factors.
4. Write the answer in simplest form.
a c ac
•

b d bd
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x  3x • x  x  2 .
x2  2x  3 x2  2x  3
2
Example: Multiply
2
x( x  3) • ( x  1)( x  2)

( x  3)( x  1) ( x  3)( x  1)
Factor the numerator and
denominator of each fraction.
x( x  3)( x  1)( x  2)

( x  3)( x  1)( x  3)( x  1)
Multiply.
x( x  3)( x  1)( x  2)

( x  3)( x  1)( x  3)( x  1)
x( x  2)

( x  1)( x  3)
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Divide by the common factors.
Write the answer in simplest
form.
4
To divide rational expressions:
1. Multiply the dividend by the reciprocal of the
a
b
divisor. The reciprocal of
is .
b
a
2. Multiply the numerators. Then multiply the
denominators.
3. Divide by the common factors.
4. Write the answer in simplest form.
a c a • d ad
 

b d b c bc
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2
2
x

x
y
2
x

2
x
y.
Example: Divide

z
z2
x  x2 y •
z2

z
2x  2x2 y
Multiply by the reciprocal
of the divisor.
x(1  xy) • z 2

2 x(1  xy) • z
Factor and multiply.
x(1  xy) • z 2

2 x(1  xy) • z
z

2
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Divide by the common
factors.
Simplest form
6
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.
10 = (5 • 2)
factors of 10
4 = (2 • 2)
LCM = 2 • 2 • 5 = 20
factors of 4
2. Find the LCM of 4x2 + 4x and x2 + 2x + 1.
4x2 + 4x = (4x)(x +1) = 2 • 2 x (x + 1)
x2 + 2x + 1 = (x +1)(x +1)
factors of x2 + 2x + 1
LCM = 2 • 2 x (x +1)(x +1) = 4x3 + 8x2 + 4x
factors of 4x2 + 4x
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Fractions can be expressed in terms of the least common multiple
of their denominators.
x
2x 1
Example: Write the fractions 2 and 2
in terms of the
4x
6 x  12 x
LCM of the denominators.
The LCM of the denominators is 12x2(x – 2).
x
x
3( x  2) 3( x  2)( x)
•


2
(2 x)( 2 x) 3( x  2) 12 x 2 ( x  2)
4x
2 x(2 x  1)
2x 1
2x 1 • 2x


2
6 x  12 x 6 x( x  2) 2 x 12 x 2 ( x  2)
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LCM
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To add rational expressions:
1. If necessary, rewrite the fractions with a common
denominator.
2. Add the numerators of each fraction.
a c ac
 
b b
b
To subtract rational expressions:
1. If necessary, rewrite the fractions with a common
denominator.
2. Subtract the numerators of each fraction.
a c ac
 
b b
b
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Example: Add 2 x  5 x .
14 14
2 x  5 x 7x x



14
14 2
2x
4
Example: Subtract 2
.
 2
x 4 x 4
2
2( x  2)
2x  4


 2
x  4 ( x  2)( x  2) ( x  2)
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Two rational expressions with different denominators can be
added or subtracted after they are rewritten with a common
denominator.
Example: Add x  3  6 .
x2  2x x2  4
x 3
6


x( x  2) ( x  2)( x  2)
x  3 • ( x  2)
6
( x)
•


x( x  2) ( x  2) ( x  2)( x  2) ( x)
( x  3)( x  2)  6 x x 2  x  6  6 x


x( x  2)( x  2)
x( x  2)( x  2)
( x  6)( x  1)
x 2  5x  6


x( x  2)( x  2) x( x  2)( x  2)
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x2
1
Example: Subtract 2
.
 2
x 1 x 1
x2  1
 2
x 1
Add numerators.
( x  1)( x  1)

( x  1)( x  1)
Factor.
( x  1)( x  1)

( x  1)( x  1)
Divide.
1
Simplest form
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