This document contains notes on rational expressions. It begins by defining a rational expression as a fraction with a polynomial in the numerator and denominator. It then discusses how to simplify rational expressions by factoring polynomials and canceling common factors. Several examples are worked through. Later, the document covers adding, subtracting, multiplying, and dividing rational expressions, providing steps and examples for each operation. It concludes by discussing complex fractions, which are fractions with fractions in the numerator or denominator, and methods for simplifying them.

1. Unit 3 Notes Rational Expressions.notebook October 31, 2012
Rational Expressions Warm Up:
May 16­12:43 PM May 16­12:43 PM
What is a rational expression?
Simplifying
Rational A rational expression is a fraction with a polynomial in the
numerator and denominator.
Expressions
May 16­12:43 PM May 16­12:43 PM
Simplifying rational expressions (fractions)
Remember that when you
divide like bases, you
SUBTRACT the
exponents!
Also, remember that
anything raised to the
0 power = 1
I kn
ow sh
it m e is
ore goin
com g
plicat to make
this ed th
! an
See next slide for answer...
May 16­1:12 PM May 16­1:12 PM
1

2. Unit 3 Notes Rational Expressions.notebook October 31, 2012
Rewrite as a
fraction
Don't use next 2 slides Break into
3 separate
fractions
Divide!
Will t
hey a
ll
be th
is
ugly?
??
Aug 1­10:29 AM May 16­2:40 PM
If the polynomial in either the numerator or
denominator is factorable, you must factor
(GCF, Difference of 2 Perfect Squares, Trinomial)
first and then simplify by canceling!
Difference
of 2 Per. Sq.!
Trinomial!
May 16­2:41 PM May 17­8:09 AM
Simplify each rational expression:
Steps for simplifying rational expressions 1. 2. Hint: Factor first!!!
(Reducing fractions)
1. Simplify any polynomial into its factored
form (GCF, DOTS, Trinomial, Arc, etc. )
3. Hint: Factor first!!! 4. Hint: Factor first!!!
2. Cancel out any factors where possible.
3. Write your final answer. 5. 6.
Hint: Factor first!!! Hint: Factor first!!!
Answers on next slide
May 17­8:09 AM May 16­2:43 PM
2

3. Unit 3 Notes Rational Expressions.notebook October 31, 2012
ers
Answ
Simplify each rational expression:
1. 2.
You pretty much know
that (x + 5) will be a
factor of the
numerator since it's
3. 4. the only thing in the
denominator. How do
you get 2x 2? How do
you get -15 when you
multiply?
Or you can do the Arc
5. 6. method!!!
May 16­2:43 PM May 16­2:41 PM
Simplify: Hmm
m...I
who w wond
er
ill be
facto a
r of
nume the
rator?
Homework:
Worksheet # 1­ 10
May 17­8:26 AM May 17­8:26 AM
Worksheet # 1­ 10 Solutions:
1. 2.
3. 4.
5. 6.
7. 8.
9. 10.
May 16­2:32 PM Oct 26­7:58 AM
3

4. Unit 3 Notes Rational Expressions.notebook October 31, 2012
Multiplying and
Dividing
Rational
Expressions
Oct 26­8:01 AM May 16­2:27 PM
How do you multiply rational expressions?
Can you multiply the following: How about this • Simplify each rational expression.
one??? Try it! • Cancel out.
• Multiply (if possible).
• Write out the remaining fraction.
1 1
You have actually been doing this for years!
Try this one:
1 1
May 17­1:32 PM May 17­11:43 AM
Give these a try: Give these a try:
1. 2. 1. 2.
May 17­1:51 PM May 17­1:51 PM
4

5. Unit 3 Notes Rational Expressions.notebook October 31, 2012
Give these a try:
1. 2.
Homework:
#11­ 16
Unit 3 Pre Test MONDAY
NWEA­ Tuesday and Wednesday
May 17­1:51 PM May 17­1:51 PM
`
Journal Entry
Answers to #11­ 16 Multiplying Rational Expressions: October 29, 2012
11. 4xy 12. 6ab2
Simplify the rational expression below:
13. 6x2z 14.
15. 16.
May 17­1:51 PM May 16­2:27 PM
Dividing Rational Expressions...
t (just 1 extra step!)
shee g
W ork Dividin
oup ing & ions
Gr ly 1. Copy, Change, Flip!!!
ss
ltip xpre (Now you are back to multiplying!!!)
on Mu ional E 2. Simplify all polynomials by factoring (if you can)
Rat
3. Cancel things out!
4. Write remaining fraction!
Try one:
You can do it...you can do it...you can do it...
May 16­2:27 PM May 17­1:59 PM
5

6. Unit 3 Notes Rational Expressions.notebook October 31, 2012
How about these. Can you divide these
rational expressions?
Here's a doozie...TRY to do it!!!
1. 2.
May 17­3:51 PM May 17­3:56 PM
Homework:
p. 177 #2, 6, 8, 14- 30
even (NO #24)
May 17­3:56 PM May 18­7:57 AM
Adding &
Quiz #4 today! Subtracting
Rational
CR #4 due Expressions
tomorrow!
May 16­2:28 PM May 16­2:28 PM
6

7. Unit 3 Notes Rational Expressions.notebook October 31, 2012
Remember back in elementary school we
used to add fractions? Can you complete the Steps to Add or Subtract Rational
following 2 problems? Expressions:
1. Get a common denominator .
2. Find "new" numerators by
multiplying.
+ ­ 3. Add/ subtract the numerators
and keep the denominator .
4. Simplify (if possible).
May 18­8:06 AM May 18­8:06 AM
Oh no...this one has variables!!! Can you subtract these fractions?
+
e
s th
at i
Wh mmon r?
co inato
om
den
May 18­8:17 AM May 21­7:58 AM
Here's the problem again...Follow the same So now I think you are ready for a more challenging problem.
Try this one.
steps you have been using since 4th grade!
Remember the steps you just used:
1. Get a common denominator .
2. Find "new" numerators by multiplying.
= 3. Add/ subtract the numerators and keep the denominator .
4. Simplify (if possible).
=
Remember the secret to
finding a common
denominator...Multiply the
2 original
denominators.
+ Answer on next slide...
May 21­7:58 AM May 18­8:12 AM
7

8. Unit 3 Notes Rational Expressions.notebook October 31, 2012
Here's the solution... Try this one...Watch out for the ‐ sign!!!
= = = =
+ = = = =
a
used or
you t
ok if enomina
Is it nt d
re e?
diffe than m
May 18­8:24 AM May 18­8:32 AM
What if there are 3 fractions???
Homework:
p. 185 #4-40 eoe
May 21­8:07 AM May 18­8:32 AM
More Rational
Expressions
Oct 17­9:49 AM May 16­2:29 PM
8

9. Unit 3 Notes Rational Expressions.notebook October 31, 2012
Yesterday, we added & subtracted fractions with
monomial
denominators. Today we will add & subtract fractions with
monomial and binomial denominators.
Steps:
1. Find a common denominator by including all factors of
each bottom.
2. Find new numerators.
3. Combine like terms.
4. Simplify, if possible.
f
ors o
fact rs as
e all o
Example #1: Includ ominat n Doe
en mo
the d ur com or. sim sn't fac
yo at plify
omin Find your numerators by , so tor or
den don you a
Answer on next slide... multiplying and then e!!! re
combine like terms.
May 21­10:13 AM May 16­2:29 PM
Here are 3 more examples for you to attempt...
Try one on your own...See how far you can get!
Remember to include all factors of the denominator
in the new denominator .
May 21­10:36 AM May 21­10:42 AM
Homework: p. 185 # 52­
62 evens
May 21­8:14 AM Oct 17­9:50 AM
9

10. Unit 3 Notes Rational Expressions.notebook October 31, 2012
More Rational
Skip this section for Expressions
2012­ 13
May 16­2:29 PM May 16­2:29 PM
Today we will add & subtract fractions with onomial and
m Example #1:
Factor each
binomial denominators that are factorable!!! denominator.
Steps:
1. Factor all denominators.
2. Find a common denominator by including all factors of
each bottom.
3. Find new numerators.
4. Combine like terms.
5. Simplify, if possible.
Take all "pieces"
Example #1: of each denominator & find
Combine like terms and
simplify your answer.
new numerators.
May 21­8:14 AM May 21­8:23 AM
How about this one:
=
What does
Who are the
the bottom
"new"
factor into? numerators?
=
What is the
common
denominator?
May 21­10:46 AM May 21­11:43 AM
10

11. Unit 3 Notes Rational Expressions.notebook October 31, 2012
Uh o
forgo h...I Last one...Explain to me how to start this problem.
t ho
factor w to
differe the
nce o
pe f 2
squarerfect
s...He
me! lp
Now finish it!
Tick tock, tick tock...
May 21­1:11 PM May 21­2:37 PM
Quiz #5 today
Homework:
p.194 #2‐ 10 even CR #5 due
tomorrow at the
Quiz tomorrow!!! beginning of
class!
May 21­10:14 AM May 16­2:30 PM
You can think of complex fractions as "stacked" fractions
Complex Fractions because they are fractions stacked on top of each other.
For example, look at the following example:
(Man this sounds hard!) Numerator
Denominator
This complex fraction is formed by the quotient of 2 fractions.
Do you remember how to divide fractions?
1 1 1
2 1 2
May 16­2:30 PM May 21­2:46 PM
11

12. Unit 3 Notes Rational Expressions.notebook October 31, 2012
What if the complex fraction looks like this:
1
1
Here's the new
What do we do? complex fraction.
Now divide and
simplify. Rewrite as 2 separate
fractions.
Copy Change Flip.
Cancel when you can.
Write final answer.
See next page...
May 22­8:08 AM May 22­10:39 AM
Example:
Example: WHY???
May 22­8:25 AM May 21­2:49 PM
Using the method we are used to....
This is another
method of
simplifying Example:
complex
fractions.
You can still do it
the same way
New complex
we did before fraction. Now use
and get the same your rules for dividing
answer. rational expressions.
Can you explain
what this person
did?
(y ­ x)(y + x)
See the next slide
if you want to
see how to do it
the way we have
been doing it...
May 22­10:43 AM May 23­7:55 AM
12

13. Unit 3 Notes Rational Expressions.notebook October 31, 2012
Example:
Homework:
p. 194 #42­ 52 evens
May 21­2:49 PM May 21­2:46 PM
Certainly, you must remember how to
Dividing Polynomials & divide polynomials, right???
Synthetic Division
May 16­2:30 PM May 22­10:52 AM
Synthetic Division:
Here is another example.
1. Start by drawing an old school long division symbol .
But how on Earth do I do this one? (See below)
I can't factor the numerator or the 2. Use only the coefficients of all terms in the
denominator . numerator. Fill in as shown in the diagram.
HELP!!! 3. Take the root (opposite sign from denominator) and
place it on the outside (as shown in diagram).
4. Always bring down the first # (as shown in the
We will use a method called Synthetic Division . diagram).
5. Begin the long division process .
Synthetic Division is a method used to divide
polynomials. Sometimes the polynomials will have
common factors, and therefore divide evenly. Other Root x3 x2 x #
times it will not divide evenly, and therefore will have a
remainder.
#
May 22­10:58 AM May 22­11:29 AM
13

14. Unit 3 Notes Rational Expressions.notebook October 31, 2012
OK so here is the problem again:
The last # is
3 1 -2 1 -6 always the
Root x3 x2 x # remainder. If
this # = 0, then
3 3 12 the polynomials
divided evenly.
Once you bring 1 1 4 6
3 1 -2 1 -6 down the first
Root x3 x2 x # #, you start
multiplying and
3 3 12 Remember the original question. You were dividing an x3 by an x.
combining, just
6 like in long
Doesn't that give you x2? Here's what you do...use the #'s as
1 1 4
division! coefficients but start from1 less than the original degree.
= 1x2+ 2x + 4, R= 6
That's great, but I still don't get what these
#'s mean...where is my answer?
May 22­11:45 AM May 22­1:13 PM
Still not quite sure about this synthetic division Give this one a try on your own. Look at the steps in
stuff? Let's try an easier one... your notebook and follow them!!!
﴾x 2 ­ 7x ­ 78﴿ ﴾x + 6﴿
-5 2 7 -15
-10 +15
2 -3 0
Since this # is 0, that means
﴾x + 5﴿ divides evenly into
﴾2x2 + 7x ­ 15﴿. You can prove
this by factoring
﴾use arc method﴿!
= 2x - 3
May 23­8:14 AM May 23­10:41 AM
= 2 1 0 0 -8
2 4 8
1 2 4 0
= x2 + 2x + 4
Since the last # is 0, the
Remember to drop 1 polynomials divide
from the original degree evenly. Check your
of the polynomial and answer by multiplying
use the #'s as ﴾x­ 2﴿﴾x 2 + 2x + 4﴿.
coefficients!
May 22­10:55 AM May 22­10:58 AM
14

15. Unit 3 Notes Rational Expressions.notebook October 31, 2012
Give this one a try on your own. Don't forget to have a Everyone can do these two...Give them a shot!
# for every "place holder."
﴾2x3 ­ 5x2 ­ 4x + 6﴿ ﴾x ­ 2﴿
﴾x 4 ­ 7x ­ 6﴿ ﴾x + 1﴿
May 22­10:58 AM May 22­10:58 AM
Solving Fractional Equations
Homework:
p. 202 #54­ 64 evens
May 22­10:52 AM May 16­2:31 PM
Wa
Remember how to solve this? 1. one. it, this i
..Do
n't s an ea
Mul I just C sy
tiply r
Why ??? oss
can
I jus
t cro
ss m
ultip
ly?
Who (or what #) is bothering you?
How can you make that # go bye bye?
May 23­10:52 AM May 23­10:55 AM
15

16. Unit 3 Notes Rational Expressions.notebook October 31, 2012
Can I ju
Can I j st cros
2. ust cro 3. s mult
iply?
ss mu What
ltiply? can I d
o to m
equatio
n muc ake this
he
solve? asier to
May 23­2:38 PM May 23­2:38 PM
Whoa..
.this o
ne is t
How m
ricky. 5.
4. any an
s
get? W wers will I
hy???
May 23­2:38 PM May 23­2:39 PM
Homework: Worksheet on
p. 209 #2- 26 eoe Fractional Equations
May 23­10:55 AM May 16­2:31 PM
16

17. Unit 3 Notes Rational Expressions.notebook October 31, 2012
Review
Exam #3 tomorrow
May 16­2:31 PM May 29­8:32 AM
17