Please CLOSE YOUR LAPTOPS,

and turn off and put away your cell phones,

and get out your note taking materials.

Coming up:

• • •

Today: (Monday, 11/25)

• Hand back/review Test 3 • Lecture on Section 7.1/2 (Rational expressions) • Daily Quiz 33

at end of hour

on material in today’s lecture

Tomorrow: (Tuesday, 11/26)

• HW 38 due on section 7.1/2 • •

Practice Quiz 9 due

• Lecture on Section 10.1 (Radical functions)

Take Weekly Quiz 9

on Sec. 7.1 and Test 3

Monday, 12/2:

• HW 39 due on section 10.1

• Daily Quiz on section 10.1

• Lecture on Section 10.2

Test 3 Results:

• Average class score after partial credit: __________ • Commonly missed questions: #_________________

Grade A A-

Grade Scale

B+ B B- C+ C C- F ≥ 920 ≥ 890 ≥ 860 ≥ 830 ≥ 800 ≥ 750 ≥ 700 ≥ 670 < 670 Points % Score ≥ 92 ≥ 89 ≥ 86 ≥ 83 ≥ 80 ≥ 75 ≥ 70 ≥ 67 < 67 If you got less than 75% on Test 1, make sure to go over your test with me or a TA sometime in the next few days. This material will be used in the next unit, and it will also be covered again on the final exam.

Teachers: You can insert screen shots of any test problems you want to go over with your students here.

Sections 7.1/2

Rational Expressions

Remember this problem from Test 3?

In the last chapter, you learned how to do this problem using long division . Today we’ll look at an alternate way to do problems like this (with no remainder) using factoring .

Rational Expressions

Rational expressions

are ratios of two polynomials, just like a rational number is a ratio of two integers.

Examples of Rational Expressions:

3

x

2  2

x

4

x

 5  4 2

x

2 4

x

 3

y

 3

xy

 4

y

2 3

x

2 4

For this last one, remember that 4 can be considered a polynomial of degree 0.

Rational expressions

can be simplified, multiplied, divided, added and subtracted using factoring methods similar to the ones we use with regular

fractions

(rational

numbers

).

•

Simplifying

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

• To do this, we need to use the

Fundamental Principle of Rational Expressions:

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

PR



P QR Q

• This is similar to what you do when you simplify a

rational number

(fraction):

Example

: Simplify 105 49

Solution

: First factor the numbers:

105 = 5*21 = 5*3*7 49 = 7*7 Next

, rewrite the ratio in its factored form: 105 = 5*3*7 49 7*7

Finally

, cancel the common factors and rewrite in simplified form: 105 = 5*3*7 = 5*3 = 15 49 7*7 7 7

Simplifying a Rational Expression:

1)

Completely factor the numerator and denominator polynomials.

2) Apply the Fundamental Principle of Rational Expressions to

cancel common factors in the numerator and denominator

.

Warning!

DO NOT multiply out the factors at the end

like you did with the numbers in a simplified fraction.

Warning 2!

Only common

FACTORS

can be canceled from the numerator and denominator. Make sure any expression you eliminate is a

factor

,

not just a term within a factor

.

Example

Simplify the following expression.

7

x

 35

x

2  5

x

 7 (

x

 5 )

x

(

x

 5 )  7

x

2

x

5

x

 7  7  2

x

5

x



/

7 

/

7  2

/

x

5

/

x

 2 5

Answer:

NO!!!!

Remember, we can only cancel entire FACTORS, not terms with factors.

2x+7 is a

factor

; it could be written as (2x+7).

The 2x and the 7 are

terms

in the factor.

The following example with numbers illustrates the error in the previous case of “bad cancelling”: Correct:  9 12  3 4

Would we get the same answer if we cancelled the 7’s first?

/ Incorrect!! /

 3 4

Problem from today’s homework:

Solve by factoring both trinomials and then canceling any common factors.

( x + 3 x - 2

Example

Simplify the following expression.

7

y



y



7

  1 (

y y

  7 7 )  

1

Revisiting this problem from Test 3:

Instead of using long division, let’s try factoring and canceling: HINT: Use a 3 – b 3 = (a – b)(a 2 + ab + b 2 )

This looks like ( ) 3

-

( ) 3 . What goes in the parentheses?

a =

x

and b =

2

.

x

3 – 8 = (

x

– 2)(

x

2 + 2x + 4).

So

x

3 (

x

– 8 = (

x

/

x

– 2) (

x

2 + 2x + 4) =

x

2 + 2x + 4 Note that this is the same answer we would have gotten using long division.

To

evaluate

a rational expression for a particular value of a variable, substitute the replacement value into the rational expression in place of that variable and simplify the result.

Example

Evaluate the following expression for

y

= -2.

y

  5  2

y

  2  2  5  (  2 )   4  7  4 7

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

y

= 5?

y

 

5



2

y

 5  2  5  5  3 0 This expression is undefined!

• 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).

Example

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

9

x

3 15

x

 2 4

x

 7  45

x

The expression is undefined when 15

x 2

+ 45 = 0.

Factoring this gives 15x(x + 3) = 0, so the expression is undefined when

x

= -3 or x = 0.

The set of numbers for which an expression is defined is called the

domain

of the expression. The domain is written in

set notation

. The

domain

for the expression in this example would be:

{ x | x ≠ 0, -3}

Example 2: Find the domain of the expression

x

2

x

2  3

x

 4 

x

 20

Solution:

Set the denominator equal to zero and solve for x.

x 2 – x – 20 = 0 (x – 5)(x + 4) = 0 (x – 5) = 0 gives x = 5, and (x + 4) = 0 gives x = -4 Therefore the domain is

{x | x ≠ 5 ,-4}

Problem from today’s homework:

0, 2,-1 To answer this question, you need to find all solutions of the equation

• • • • • •

obtained by setting the denominator equal to zero.

(Notice that in

DOMAIN

questions, you focus only on the

DENOMINATOR

).

How many solutions are you expecting to find? Why?

Answer: At most

three

, because the polynomial is

cubic

, i.e. has a

degree of three

.

Find the answers by factoring the polynomial and setting each of the three factors to equal to zero.

Factoring: x(x - 2)(x + 1) Solutions: x = 0, x = 2, x = -1. These are the numbers that are NOT in the domain.

Check: Plug each of these three numbers back into the denominator of the function and show that each one gives zero as the result.

Steps in multiplying two or more rational expressions:

1. Factor all numerators and denominators.

2. Cancel any common factors between all numerators and all denominators.

3. Don’t

multiply out any polynomial terms in the answer – just leave them factored.

IMPORTANT: Always factor and cancel first instead of multiplying first, even though the directions say “multiply”.

Example

Multiply:  (

x

 2) 2 10  2

x

5  4  (

x

 2)(

x x

2)  (

x

 2) (

x x

 2) 

x

 2 4

When

dividing

rational expressions, first change the division into a

multiplication

problem, where you use the reciprocal of the divisor as the second part of the product.

Then treat it as a multiplication problem (factor, multiply, simplify).

Example

Divide the following rational expression.

(

x

 3 ) 2 5  5

x

 15 25  (

x

 3 ) 2 5  25 5

x

 15  (

x

 3 )(

x

 3 )  5  5 5  5 (

x

 3 ) 

x

 3

REMINDER:

The assignment on today’s material

(HW 38)

is due at the start of the next class session.

Please open your laptops and start working on the homework assignment if there’s any time left before it’s time to take the quiz at the end of the class session.

You can keep your yellow formula sheets out when you take the quiz.

Lab hours in 203: Mondays through Thursdays 8:00 a.m. to 7:30 p.m.

Please remember to sign in on the Math 110 clipboard by the front door of the lab

Please open Daily Quiz 33.

If you have any time left after finishing the quiz problems, CHECK YOUR FACTORING ANSWERS before you submit the quiz.

• • A scientific calculator may be used on this quiz. Remember to turn in your answer sheet to the TA when the quiz time is up.