Math 20-1 Chapter 6 Rational Expressions and Equations

6.1 Rational Expressions

Teacher Notes

Rational Number: Q ab

| a,b I ,b 0

ab

Numerator indicates how many you have

Denominator indicates what you have

23

70

6.1.1

Why can’t the denominator be zero?

Reduce a fraction to lowest terms by dividing out common factors from both the numerator and the denominator.

48

1 4 2 4

11

12

1424

12

44

1424

56

16y12y

44y34y

43

1423

1423

55

2

1

6.1.2

Cannot divide out common factors

2 32

52

4 82

2 4

12(2 4)

2

Cannot reduce with + or -between terms.

Factor first.

Reduce common factors in the numerator and denominator.

520 5

1

4 15

5(4 1)

6.1.3

6.1 Rational Expressions

Examples of Rational Expressions

54423 2

xxx

22 43234

yxyxyx

4

3 2x

Rational expressions are algebraic fractions of

the form , where P(x) and Q(x) are

polynomials and Q(x) does not equal zero.

( )( )

P xQ x

Not Rationals2x2 x

3 x1 5 2x2 3

6.1.4

A rational expression has a numerator and a denominator.While the numerator can have any value, the denominator cannot be zero.A variable value that makes the denominator zero is an excluded value or non-permissible value (NPV) of that variable.

Non-Permissible Values

Consider the rational expression: 42

xx

Set the denominator not equal to zero gives:

The non-permissible value for x is - 2. The rational expression is defined for all real numbers except -2.

x + 2 ≠ 0x ≠ - 2

4 22

,x xx

6.1.5

Determine all non-permissible values for each rational expression.

Non-Permissible Values

2

3a) 4 21

x

x x

Determine the values for which x2 + 4x – 21 ≠ 0.

(x + 7)(x – 3) ≠ 0x ≠ –7 or x ≠ 3

The non-permissible values for x are –7 and 3. 2

3 , 7,34 21

x x

x x

23 2

b) x yx y

Determine the values for which 3x – 2y ≠ 0.

2 2 3, ,3 2 3 2

x y y xx y

x y

2 33 2

and y xx y

6.1.6

Your TurnDetermine all non-permissible values.

32 8

a) mm 2

2 59 20

b) x

x x 2

79

c) pp

32 8

a) mm

NPV for m is -4

3 42 8

,m m

m

2

2 59 20

b) x

x x

NPVs for x are -4, -5

2

2 5 4 59 20

, ,x xx x

2

79

c) pp

All values of p are allowed. The domain of the expression is all real numbers

6.1.7

A rational expression is simplified or reduced to lowest terms when the numerator and denominator have no common factors other than 1.

Simplifying Rational Expressions

Examples:924

a)

9 3 324 3 8

( )( )( )( )

38

1

1

3 2

b) a babc

3 2

( )( )( )( )( )a b a a a b b

abc abc2

0 0 0 , , ,a b a b cc

1

1

1

1

6.1.8

Simplifying Rational Expressions

1. Factor both the numerator and denominator as completely as possible.

2. Divide out any factors common to both the numerator and denominator.

2

2

4 9a) 2 3

xx x

Simplify each rational expression, stating non-permissible values.

(2 3)(2 3)(2 3)( 1)

x xx x

1

1

(2 3) 3, , 1( 1) 2

x xx

NPVs:3 12

x x and

6.1.9

Simplifying Rational Expressions

2

2

20b) 25

x xx

( 5)( 4)(5 )(5 )

x x

x x1

4 , 5, 55

x xx

NPVs: 5 5 x x and –1

1( 4)(5 )

x

x

Your TurnSimplify and state all non-permissible values.

2

2

10 24a) 5 4

y yy y

6 , 1,41

y yy

2

2

16b) 7 12

x

x x

4 , 3, 43

x xx

6.1.10

AssignmentSuggested Questions:

Page 317:4, 6, 7, 8a,c,e, 9, 11, 13, 15, 19, 20, 22,

6.1.11

Write a rational expression equivalent towith a denominator of 6x(x+2)

12xx