7.1: Simplifying Rational Expressions

March 31, 2009

Topics for review

• Multiplying monomials• Graphing• Factoring

Multiplying monomials

Graphing

Graphing

factoring

• Find ac and b• Find two numbers that add to equal b and

multiply to equal ac• Split b in half, using these numbers• Factor by grouping

Factoring

5x2-8x-4=0

Factoring

9x2+4x-5=0

This week

• Monday: review• Tuesday: lecture (7.1)• Wednesday: work day• Thursday: lecture (7.2)• Friday: quiz (7.1 only)/work day

Objectives

• Simplify rational expressions• Identify rational functions• Simplify rational functions• Graph rational functions

Standards Addressed

• Algebraic Relationships– Analyze the nature of change of each variable in a

non-linear relationship as suggested by a table of values, a graph or a formula

– Evaluate and make a table for two-variable formulas and match a graph or table of values to its formula

• Calculations and Estimations– Apply the associative, commutative, and distributive

properties to simplify computations with real numbers

What does rational mean?

• Rational comes from the word ratio, which means

fraction!• Rational expressions are expressions that

can be written as fractions.

Precluding division by zero

Any value divided by zero is undefined.

Find the value for x for which the following rational expressions are undefined.

1

103

x

x

21

2

10

x

A few more

1021

57

x

x

2

53 x

Evaluating a rational expression

To evaluate means to solve for a given value of x.

Evaluate the following rational expression for x = 3.

52

3

xx

A few more

• Evaluate for x= -2.

23

62

x

xx

x

2

36

Simplifying a rational expression

• Monomial: one term• Simplifying monomials:

Cancel out common factors

310

125

8

6

yx

yx

A couple more

17312

15102

24

56

zyx

zyx

126

64

60

36

ba

ba

Simplying non-monomial rational expressions1) Factor the numerator and the

denominator, using the GCF or the ac method (sometimes factoring out -1 can be helpful)

2) Cancel out common factors

An example

Try this one:

44

22

2

xx

xx

Try another

24

42

x

x

And another…

32

622

2

xx

xx

Identifying a rational function

• A rational function must be able to be written as a ratio, even if the denominator is simply 1.

• There cannot be any square roots.

Simplifying a rational function

1) Examine the denominator. Determine what value of x will make the denominator equal zero. Write x ≠ (that number).

2) Then simplify as before.

3) The answer will be the simplify version, plus part 1.

Try it

• Try this one.

1

12)(

2

x

xxxf

Try another

x

xxxf

5

105)(

2

Graphing a rational function

1) Simplify the function.

2) Note what x cannot equal (≠).

3) Plug that value in and determine the y value. Mark an open circle at the ordered pair (x, y) that you have just found. This is called a hole.

4) Graph normally.

Try it

Remember this one?

1

12)(

2

x

xxxf

Try another

What about this one?

x

xxxf

5

105)(

2

Your assignment

• Pages 489-490

2-36 even

40-66 even