Rational Expressions
• Fundamentals of College Algebra
Rational Expressions
•
•
•
Properties of
Rational
Expressions
Arithmetic
Operations on
Rational
Expressions
Methods for
Simplifying
Complex
Fractions
• Simplifying a Rational Expressions
• Dividing a Rational Expression
• Determining the LCD of Rational
Expressions
• Adding and Subtracting Rational
Expressions
• Simplifying Complex Fractions
• Solving An Application
Rational Expressions
Properties of Rational Expressions
A rational
expression is a
fraction in which
the numerator
and denominator
are polynomials
For all rational expressions P/Q and R/S where Q ≠ 0
and S ≠ 0,
Equality
P/Q = R/S iff PS = QR
Equivalent Expressions
P/Q = PR/QR, R ≠ 0
Sign
- P/Q = (-P)/Q = P/(-Q)
Rational Expressions
To simplify a
rational
expression,
factor the
numerator and
denominator.
Then eliminate
any common
factors.
7 + 20x – 3x²
2x² - 11x – 21
=
=
=
-
- (3x² - 20x – 7)
2x² - 11x – 21
- (3x + 1)(x – 7)
(2x + 3)(x – 7)
3x + 1
2x + 3
Restrictions: x ≠ 7 and x ≠ 3/2
Rational Expressions
For all rational expressions P/Q, R/Q, and R/S where
Q ≠ 0 and S ≠ 0,
Arithmetic
Operations
Defined on
Rational
Expressions
Addition
P/Q + R/Q = (P + R)/Q
Subtraction
P/Q – R/Q = (P – R)/Q
Multiplication
P/Q ∙ R/S = (PR)/(QS)
Division
P/Q ÷ R/S
= P/Q ∙ S/R =
R≠0
PS/QR
Rational Expressions
Divide A Rational Expression
When
multiplying or
dividing rational
expressions,
factor and
eliminate any
like terms
x² + 6x + 9 ÷ x² + 7x + 12
x³ + 27
x³ - 3x² + 9x
(x + 3)²
÷ (x + 4)(x + 3)
(x + 3)(x² - 3x + 9)
x(x² - 3x + 9)
(x + 3)(x + 3)
∙ x(x² - 3x + 9)
(x + 3)(x² - 3x +9)
(x + 4)(x + 3)
x
x+4
Rational Expressions
Determining the
LCD of Rational
Expressions
1. Factor each denominator completely
2. Express repeated factors with exponential
notation
3. Identify the largest power of each factor in any
single factorization
4. The LCD is the product of each factor raised to
its largest power.
1
x+3
5x
(x + 5)(x – 7)³
and
and
5
2x + 1
have an LCD of (x +3)(2x + 1)
7
x(x + 5)²(x - 7)
have an LCD of
x(x + 5)²(x – 7)³
Rational Expressions
5x + x
48
15
Add and
Subtract
Rational
Expressions
48 = 24 ∙ 3
24
15 = 3 ∙ 5
∙ 3 ∙ 5 = 240
5x ∙ 5 + x ∙ 16
48 ∙ 5
15 ∙ 16
25x + 16x
240
240
41x
240
Find the prime factorization
of the denominators
LCD is the product of each
factor raised to its highest
power
Rational Expressions
x
x² - 4
Adding and
Subtracting
Rational
Expressions
-
2x – 1
x² - 3x - 10
x² - 4 = (x + 2)(x – 2)
x² - 3x – 10 = (x + 2)(x – 5)
LCD is the product each
factor raised to its highest
power
(x + 2)(x – 2)(x – 5)
x(x – 5)
(x + 2)(x – 2)(x – 5)
x(x – 5) – (2x – 1)(x – 2)
(x + 2)(x – 2)(x – 5)
=
Factor the denominators
completely
(2x – 1)(x – 2)
(x + 2)(x – 2)(x –5)
=
x² - 5x – (2x² - 5x + 2)
(x + 2)(x – 2)(x – 5)
x² - 5x – 2x² + 5x - 2
-x² - 2
=
(x + 2)(x – 2)(x – 5)
(x + 2)(x – 2)(x – 5)
Rational Expressions
Method 1: Multiply by the LCD
1.
Methods for
Simplifying
Complex
Fractions
2.
3.
Determine the LCD of all the fractions in the
complex fraction
Multiply both the numerator and
denominator of the complex fraction by the
LCD.
If possible, simplify the resulting rational
expression.
Method 2: Multiply by the reciprocal of the
denominator
1. Simplify the numerator and denominator to
single fractions.
2. Multiply the numerator by the reciprocal of the
denominator
3. If possible, simplify the resulting rational
expression.
Rational Expressions
3- 2
a
Simplify A
Complex
Fraction
1+
4
a
Multiply by the LCD of
all the fractions and
then simplify
Rational Expressions
3- 2
a
1+
4
a
Simplify the numerator and
denominator to single
fractions then multiply by
the reciprocal of the
denominator.
Rational Expressions
2
x-2
Simplify A
Complex
Fraction
3x
x-5
+
-
1
x
2
x-5
Simplify the numerator and
denominator to single
fractions then multiply by
the reciprocal of the
denominator.
2
x-2
Simplify A
Complex
Fraction
3x
x-5
+
-
1
x
2
x-5
Multiply by the LCD of
all the fractions and
then simplify
Rational Expressions
c-1
a-1 + b-1
Simplify A
Complex
Fraction
Rational Expressions
Solve
An
Application
The average speed for a roundtrip is given by the
complex fraction:
2
1 + 1
v1 v2
Find the average speed for a round trip when
v1 = 50 mph and v2 = 40 mph.
Rational Expressions
Assignment
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