Rational Expressions

Martin-Gay, Developmental Mathematics 2

Rational Expressions

Examples of Rational Expressions

54

423 2

x

xx22 432

34

yxyx

yx

4

3 2x

Martin-Gay, Developmental Mathematics 3

To evaluate a rational expression for a particular value(s), substitute the replacement value(s) into the rational expression and simplify the result.

Evaluating Rational Expressions

Example

Evaluate the following expression for y = 2.

y

y

5

2)

2 2( 25

7

4

7

4

Martin-Gay, Developmental Mathematics 4

In the previous example, what would happen if we tried to evaluate the rational expression for y = 5?

y

y

5

2 5 25 5 0

3

This expression is undefined!

Evaluating Rational Expressions

Martin-Gay, Developmental Mathematics 5

We have to be able to determine when a rational expression is undefined.

A rational expression is undefined when the denominator is equal to zero.

The numerator being equal to zero is okay (the rational expression simply equals zero).

Undefined Rational Expressions

Martin-Gay, Developmental Mathematics 6

Find any real numbers that make the following rational expression undefined.

4515

49 3

x

xx

The expression is undefined when 15x + 45 = 0.

So the expression is undefined when x = 3.

Undefined Rational Expressions

Example

Simplifying Rational Expressions

Martin-Gay, Developmental Mathematics 8

Simplifying a rational expression means writing it in lowest terms or simplest form.

If P, Q, and R are polynomials, and Q and R are not 0,

Q

P

QR

PR

Simplifying Rational Expressions

Martin-Gay, Developmental Mathematics 9

Simplifying Rational Expressions

A “rational expression” is the quotient of two polynomials. (division)

2

2

nn

23

1032

x

xx

Martin-Gay, Developmental Mathematics 10

Simplifying Rational Expressions

A rational expression is in simplest form when the numerator and denominator have no common factors (other than 1)

15

9Simplify

53

33

5

3

Martin-Gay, Developmental Mathematics 11

Simplifying Rational Expressions

53

33Simplify

Martin-Gay, Developmental Mathematics 12

How to get a rational expression in simplest form…

Factor the numerator completely (factor out a common factor)

Factor the denominator completely (factor out a common factor)

Cancel out any common factors (not addends)

Martin-Gay, Developmental Mathematics 13

Difference between a factor and an addend

A factor is in between a multiplication sign

An addend is between an addition or subtraction sign

Example:

x + 3 3x + 9

x – 9 6x + 3

3( 3)

3(2 1)

x

x

3( 3)

3(2 1)

x

x

Martin-Gay, Developmental Mathematics 14

x

x

5

105 :Simplify

x

x

5

25

Factor

x

x 2

x

x 2

Martin-Gay, Developmental Mathematics 15

1

23 :Simplify

2

2

y

yy

11

21

yy

yy1

2

y

y

Martin-Gay, Developmental Mathematics 16

y

y

48

2412 :Simplify

yy

412

212

y

y

4

2

Martin-Gay, Developmental Mathematics 17

xx

xx

23

2 :Simplify

2

2

2 1

3 2

x x

x x

2 1

3 2

x

x

Martin-Gay, Developmental Mathematics 18

2

2

9Simplify:

2 15

a

a a

3 3

5 3

a a

a a

( 3)

( 5)

a

a

Martin-Gay, Developmental Mathematics 19

x

x

4

4 :Simplify

41

4

x

x

11

1

Martin-Gay, Developmental Mathematics 20

Wentz’s Shortcut

1a b

b a

Martin-Gay, Developmental Mathematics 21

Opportunity for Mathematical Growth

Please do your handout (17 questions)