Upload
ariel-harris
View
222
Download
0
Embed Size (px)
Citation preview
Rational Expressions
Rational Expressions
• Fundamentals of College Algebra• Fundamentals of College Algebra
Rational ExpressionsRational 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 ExpressionsRational 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 ExpressionsRational Expressions
7 + 20x – 3x²2x² - 11x – 21
= - (3x² - 20x – 7) 2x² - 11x – 21= - (3x + 1)(x – 7) (2x + 3)(x – 7)= 3x + 1
2x + 3
To simplify a rational
expression, factor the
numerator and denominator.
Then eliminate any common
factors.
-
Restrictions: x ≠ 7 and x ≠ 3/2
Rational ExpressionsRational ExpressionsFor all rational expressions P/Q, R/Q, and R/S
where Q ≠ 0 and S ≠ 0,
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 = PS/QR
R ≠ 0
Arithmetic Operations Defined on
Rational Expressions
Rational ExpressionsRational 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 ExpressionsRational Expressions1. Factor each denominator completely2. Express repeated factors with exponential
notation3. Identify the largest power of each factor
in any single factorization4. The LCD is the product of each factor
raised to its largest power.
Determining the LCD of Rational
Expressions
1 x + 3
and 5 2x + 1
have an LCD of (x +3)(2x + 1)
5x (x + 5)(x – 7)³
and 7 x(x + 5)²(x - 7)
have an LCD of
x(x + 5)²(x – 7)³
Rational ExpressionsRational Expressions
Add and Subtract Rational
Expressions
5x48
+ x 15
Find the prime factorization of the denominators48 = 24 ∙ 3 15 = 3 ∙ 5
LCD is the product of each factor raised to its highest power
24 ∙ 3 ∙ 5 = 240
5x ∙ 548 ∙ 5
+ x ∙ 16 15 ∙ 16
25x 240
+ 16x240
41x240
Rational ExpressionsRational Expressions
Adding and Subtracting
Rational Expressions
x x² - 4
- 2x – 1 x² - 3x - 10
Factor the denominators completely
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) -
(2x – 1)(x – 2) (x + 2)(x – 2)(x –5)
x(x – 5) – (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 (x + 2)(x – 2)(x – 5) =
-x² - 2(x + 2)(x – 2)(x – 5)
Rational ExpressionsRational Expressions
Methods for Simplifying
Complex Fractions
Method 1: Multiply by the LCD
1. Determine the LCD of all the fractions in the complex fraction
2. Multiply both the numerator and denominator of the complex fraction by the LCD.
3. 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 ExpressionsRational Expressions
Simplify A Complex Fraction
3 - 2a
1 +4a
Multiply by the LCD of all the fractions and then simplify
Rational ExpressionsRational Expressions
3 - 2a
1 +4a
Simplify the numerator and denominator to single fractions then multiply by the reciprocal of the denominator.
Rational ExpressionsRational Expressions
Simplify A Complex Fraction
2 x - 2 +
1x
3x x - 5
- 2 x - 5
Simplify the numerator and denominator to single fractions then multiply by the reciprocal of the denominator.
2 x - 2 +
1x
3x x - 5
- 2 x - 5
Multiply by the LCD of all the fractions and then simplify
Simplify A Complex Fraction
Rational ExpressionsRational Expressions
Simplify A Complex Fraction
c-1
a-1 + b-1
Rational ExpressionsRational Expressions
Solve An
Application
The average speed for a roundtrip is given by the complex fraction:
2 1 v1
+ 1 v2
Find the average speed for a round trip when v1 = 50 mph and v2 = 40 mph.
Rational ExpressionsRational ExpressionsAssignment
Page 45 – 46# 1 – 57 odd