Rational Expressions Fundamentals of College Algebra.

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


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)


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



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


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


Adding and Subtracting


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)


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.


Simplify A Complex Fraction

3 - 2a

1 +4a

Multiply by the LCD of all the fractions and then simplify


3 - 2a

1 +4a

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

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




c-1

a-1 + b-1


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

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Rational Expressions Fundamentals of College Algebra.

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