1

P.5 RATIONAL EXPRESSIOS

Objectives:

Simplify Rational Expression

Operations on Rational Expressions

Determining the LCD of Rational Expressions

Complex Fractions

التعبير )(الكسرية

2

Polynomial – The sum or difference of monomials. Rational expression – A fraction whose numerator

and denominator are polynomials. Domain of a rational expression – the set of all real

numbers except those for which the denominator is zero.

Reduced form – a rational expression in which the numerator and denominator have no factors in common.

3

Rational Function:

Let p(x) and q(x) be polynomials. Then a rational function is given by:

p xf x

q x

The domain (values x can take) of the fcn includes all x-values such that q(x) 0.

4

Ex: Find the domain:

The domain of f is all values of x, … but x2 - 4 0; solve by factoring … (x - 2)(x + 2) 0; By ZPP, x-2 0; x+2 0 x 2

2

2

5. ( )

4

xa f x

x

, 2 2,2 2, Domain=

5

solve: x2 + 1 0

x2 -1

No real solutions …. There are no restrictions on the values x can take.

2

4. ( )

1b f x

x

1x

Domain = ,

6

Rational Expressions

P, Q, R, and S are polynomials

Addition

Operation

Multiplication

Subtraction

Division

P Q P Q

R R R

P Q P Q

R R R

P Q PQ

R S RS

P Q P S PS

R S R Q RQ

Notice the common denominator

Find the reciprocal and multiply

7

Cancellation Properties

ac bc a b c implies if 0

acbc

ab

b c if 0 0,

8

Rational Expressions

Ex: Simplify

2

2

25

7 10

x

x x

5 5

2 5

x x

x x

Cancel common factorsFactor

Ex: Simplify

2 2

3 2

2 1 6 6

1

x x x x

x x

3

1 1 6 1

1 1

x x x x

x xx

FactorCancel common factors

Multiply Across

5

2

x

x

2

6 1x

x

11

Least Common Denominator( LCD)

LCD P LCD RP R Q SQ S LCD

product of all prime factors in and

raised to the highest power.

LCD P Q

المشترك ) المقام(االصغر

12

Adding/Subtracting

Ex: Simplif4

3 2

yx x

Combine like terms

3 4 2

( 4) 4

x x

x x x x

Must have LCD: x(x + 4)

3 12 2 5 12

( 4) 4

x x x

x x x x

Distribute and combine fractions

13

Complex Fractions

Ex: Simplify

32

9

4 x

xx

Factor to get hereDistribute and reduce to get here

32

94

xx

x xx

2

3 2

9 4

x

x

Multiply by the LCD: x

3 2 1

3 2 3 2 3 2

x

x x x

المركبة ) (الكسور

14

Ex: Simplify

12 12 3

2 3 2

2 2

9 3 9 27

64 5 24

a a a a

a a a

21 2 3 11 4

3 1 5

x x

x x x

1 1

2 2

xy yx

x y