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RATIONAL
EXPRESSIONS
Definition of a Rational Expression
A rational number is defined as the ratio of two integers, where q ≠ 0
Examples of rational numbers:
p
q
2 1, ,93 5
A rational expression is defined as the ratio of two polynomials, where q ≠ 0.
Examples of rational expressions:
p
q
2
2
3 6 3 6 2, ,4 4 7
x r r
x
Domain of a Rational Expression
Domain of a Rational ExpressionThe domain of a Rational Expression is the set of all real numbers that when substituted into the Expression produces a real number.
3
2
x
x
If you choose x = 2, the denominator will be 2 – 2 = 0 which is illegal because you can't divide by zero. The answer then is:
{x | x 2}.
illegal if this is zero
Note: There is nothing wrong with the top = 0 just
means the fraction = 0
Finding the Domain of Rational Expression
2
10
25
a
a
Set the denominator equal to zero. The equation is quadratic.
2 25 0a Factor the equation
( 5)( 5) 0a a Set each factor equal to zero.Solve
5 0a 5 0a The domain is the set of real numbers except 5 and -5
Domain: {a | a is a real number and a ≠ 5, a ≠ -5}
5a 5a
2
2 14
49
p
p
2( 7)p
( 7)( 7)p p
REDUCING RATIONAL EXPRESSIONS
To reduce this rational expression, first factor the numerator and the denominator.
To find the domain restrictions, set the denominator equal to zero. The equation is quadratic.
( 7)( 7) 0p p Set each factor equal to 0.
7 0p 7 0p
p = -7 or p = 7
The domain is all real numbers except -7 and 7.
2
2 14
49
p
p
CAUTION:Remember when you
have more than one term, you cannot cancel with
one term. You can cancel factors only.
2( 7)p
( 7)( 7)p p
2
7p
REDUCING RATIONAL EXPRESSIONS
There is a common factor so we can reduce.
Provided p ≠ 7 and p ≠ -7
2
2 8
10 80 160
c
c c
To reduce this rational expression, first factor the numerator and the denominator.
CAUTION:Remember when you
have more than one term, you cannot cancel with
one term. You can cancel factors only.
2( 4)c
2 5( 4)( 4)c c
1
5( 4)c
REDUCING RATIONAL EXPRESSIONS
There is a common factor so we can reduce.
1
Simplifying a Ratio of -1
51
5
The ratio of a number and its opposite is -1
21
2
x
x
2 1( 2 ) 1( 2) 1
12 ( 2) ( 2) 1
x x x
x x x
factor out a -1
3 3c d
d c
Simplifying Rational Expressions to Lowest Terms.
To reduce this rational expression, first factor the numerator and the denominator.
Reduce common factors to lowest terms.
33
1
3( )c d
Notice that (c – d) and (d – c) are opposites and form a ratio of -1
1( )c d
Solution
2
5
25
y
y
Simplifying Rational Expressions to Lowest Terms.
To reduce this rational expression, first factor the numerator and the denominator.
Reduce common factors to lowest terms.
1
5y
1( 5)y
Notice that (y - 5) and (5 – y) are opposites and form a ratio of -1
( 5)( 5)y y
Solution
Multiplication of Rational Expressions
Multiplication of Rational ExpressionsLet p, q, r, and x represent polynomials, such that q ≠ 0 s ≠ 0. Then,
p r pr
q s qs
2 2
3 3 2
6
c d
c c d
To multiply rational expressions we multiply the numerators and then the denominators. However, if we can reduce, we’ll want to do that before combining so we’ll again factor first.
3( )c d
2 3 c
1
( )c c d
MULTIPLYING RATIONAL EXPRESSIONS
c d c d
Now cancel any like factors on top with any like factors on bottom.
Simplify.
1
1
Division of Rational Expressions
Division of Rational ExpressionsLet p, q, r, and x represent polynomials, such that q ≠ 0 s ≠ 0. Then,
p r p s ps
q s q r qr
2
5 5 10
2 9
t
t
To divide rational expressions remember that we multiply by the reciprocal of the divisor (invert and multiply). Then the problem becomes a multiplying rational expressions problem.
5( 3)t
25
3t
DIVIDING RATIONAL EXPRESSIONS
2 5
( 3)( 3)t t
2
5 1529
10
t
t
Multiply by reciprocal of bottom fraction.
1
To divide rational expressions remember that we multiply by the reciprocal of the divisor (invert and multiply). Then the problem becomes a multiplying rational expressions problem.
DIVIDING RATIONAL EXPRESSIONS
2 2
2
11 30 30 5
10 250 2 4
p p p p
p p
Multiply by reciprocal of bottom fraction.
2
2 2
11 30 2 4
10 250 30 5
p p p
p p p
Factor
( 5)( 6)p p 2( 2)p
2 5( 5)( 5)p p 5 (6 )p p
Notice that (p - 6) and (6 – p) are opposites and form a ratio of -1
5 ( 6)p p Reduce common factors
( 2)
25 ( 5)
p
p p
Solution
Addition and Subtraction of Rational Expressions
Addition and Subtraction of Rational Expressions
Let p, q, and r represent polynomials where q ≠ 0. Then,
1.
2.
Addition and Subtraction of Rational Expressions
Let p, q, and r represent polynomials where q ≠ 0. Then,
1.
2.
p r p r
q q q
p r p r
q q q
Adding and Subtracting Rational Expressions with a Common
Denominator
1 7
12 12 The fractions have the same denominator.
Add term in the numerators, and write the result over the common denominator.
1 7
12
8
12
Simplify to lowest terms.
2
3 Solution
Adding and Subtracting Rational Expressions with a Common
Denominator2 5 24
3 3
x x
x x
The fractions have the same denominator.
Subtract the terms in the numerators, and write the result over the common denominator.
2 5 24
3
x x
x
Simplify the numerator.
Factor the numerator and denominator to determine if the rational expression can be simplified.
2 5 24
3
x x
x
( 8)( 3)
3
x x
x
Simplify to lowest terms.8x
Steps to Add or Subtract Rational Expressions
• Factor the denominators of each rational expression.
• Identify the LCD• Rewrite each rational expression as an
equivalent expression with the LCD as its denominator.
• Add or subtract the numerators, and write the result over the common denominator.
• Simplify to lowest terms.
To add rational expressions, you must have a common denominator. Factor any denominators to help in determining the lowest common denominator.
ADDING RATIONAL EXPRESSIONS
2
1 10
5 25x x
5 5x x ( 5)x
So the common denominator needs each of these factors.
This fraction needs (x + 5)
This fraction needs nothing
101
5
5
5x x
x
5
5 5
x
x x
simplifydistribute
Reduce common factors1
1
( 5)x Solution
5 10
5 5
x
x x
To add rational expressions, you must have a common denominator. Factor any denominators to help in determining the lowest common denominator.
ADDING RATIONAL EXPRESSIONS
1 5
7 7d d
The expressions d - 7 and 7- d are opposites and differ by a factor of -1 Therefore, multiply the numerator and denominator of either expression by -1 to obtain a common denominator.
1 ( 5)
7d
Simplify
Solution
1 5( 1)
( 1)7 (7 )d d
4
7d
Add the terms in the numerators, and write the result over the common denominator.
To subtract rational expressions, you must have a common denominator. Factor any denominators to help in determining the lowest common denominator.
SUBTRACTING RATIONAL EXPRESSIONS
2 4 1
3 2
q q
23
So the common denominator needs each of these factors.The LCD is 6.
This fraction needs (2)
This fraction needs (3)
2 3(2 14 )
6
11
6
q
simplifydistribute
Reduce common factors
Solution
4 8 3 3
6
q q
Subtracting rational expressions is much like adding, you must have a common denominator. The important thing to remember is that you must subtract each term of the second rational function.
SUBTRACTING RATIONAL EXPRESSIONS
2
2 2
3
4 12 4
x x
x x x
6 2x x 2 2x x
So a common denominator needs each of these factors.
6 2 2x x x
This fraction needs (x + 2)
This fraction needs (x + 6)
2
6 2
6
2
2 3x x
x
x
x
x
x
-
3 2 22 9 18
6 2 2
x x x x
x x x
3 2 9 18
6 2 2
x x x
x x x
Distribute the negative to each term.
FOIL
2 9 18x x