Unit 3 notes rational expressions

Unit 3 Notes Rational Expressions.notebook

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October 31, 2012

May 1612:43 PM

Rational Expressions

May 1612:43 PM

Warm Up:

May 1612:43 PM

SimplifyingRational Expressions

May 1612:43 PM

What is a rational expression?

A rational expression is a fraction with a polynomial in the numerator and denominator.

May 161:12 PM

Simplifying rational expressions (fractions)

I know she is going to make

it more complicated than this!

See next slide for answer...

May 161:12 PM

Remember that when you divide like bases, you

SUBTRACT the exponents!

Also, remember that anything raised to the

0 power = 1


2

October 31, 2012

Aug 110:29 AM

Don't use next 2 slides

May 162:40 PM

Rewrite as a fraction

Break into 3 separate fractions

Divide!

Will they all be this ugly???

May 162:41 PM May 178:09 AM

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 178:09 AM

Steps for simplifying rational expressions (Reducing fractions)

1. Simplify any polynomial into its factored form (GCF, DOTS, Trinomial, Arc, etc.)

2. Cancel out any factors where possible.

3. Write your final answer.

May 162:43 PM

Simplify each rational expression:

1. 2.

3. 4.

5. 6.

Hint: Factor first!!!





Answers on next slide


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October 31, 2012

May 162:43 PM

Simplify each rational expression:

1. 2.

3. 4.

5. 6.

Answers

May 162:41 PM

You pretty much know that (x + 5) will be a

factor of the numerator since it's the only thing in the

denominator. How do you get 2x2? How do you get -15 when you

multiply?

Or you can do the Arc method!!!

May 178:26 AM

Hmmm...I wonder who will be a factor of the numerator?

Simplify:

May 178:26 AM

Homework:Worksheet # 1 10

May 162:32 PM

Worksheet # 1 10 Solutions:

1. 2.

3. 4.

5. 6.

7. 8.

9. 10.

Oct 267:58 AM


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October 31, 2012

Oct 268:01 AM May 162:27 PM

Multiplying and Dividing Rational

Expressions

May 171:32 PM

How do you multiply rational expressions?Can you multiply the following:

You have actually been doing this for years!

Try this one:

May 1711:43 AM

• Simplify each rational expression.• Cancel out.

• Multiply (if possible).• Write out the remaining fraction.

How about this one??? Try it!

1

1

1

1

May 171:51 PM

Give these a try:

1. 2.

May 171:51 PM

Give these a try:

1. 2.


5

October 31, 2012

May 171:51 PM

Give these a try:

1. 2.

May 171:51 PM

Homework:#11 16

Unit 3 Pre Test MONDAY

NWEA Tuesday and Wednesday

May 171:51 PM

Answers to #11 16 Multiplying Rational Expressions:

11. 4xy 12. 6ab2

13. 6x2z 14.

15. 16.

May 162:27 PM

Journal Entry October 29, 2012

Simplify the rational expression below:

`

May 162:27 PM

Group Worksheet

on Multiplying & Dividing


May 171:59 PM

Try one:

Dividing Rational Expressions...(just 1 extra step!)

1. Copy, Change, Flip!!!(Now you are back to multiplying!!!)

2. Simplify all polynomials by factoring (if you can)3. Cancel things out!4. Write remaining fraction!

You can do it...you can do it...you can do it...


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October 31, 2012

May 173:51 PM

How about these. Can you divide these rational expressions?

1. 2.

May 173:56 PM

Here's a doozie...TRY to do it!!!

May 173:56 PM

Homework:p. 177 #2, 6, 8, 14- 30 even (NO #24)

May 187:57 AM

May 162:28 PM

Quiz #4 today!CR #4 due tomorrow!

May 162:28 PM

Adding & Subtracting



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October 31, 2012

May 188:06 AM

+

Remember back in elementary school we used to add fractions? Can you complete the

following 2 problems?

May 188:06 AM

Steps to Add or Subtract Rational 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 188:17 AM

+

Oh no...this one has variables!!!

May 217:58 AM

Can you subtract these fractions?

What is the

common

denominator?

May 217:58 AM

=

=

Here's the problem again...Follow the same steps you have been using since 4th grade!

Remember the secret to finding a common

denominator...Multiply the 2 original

denominators.

May 188:12 AM

+

So now I think you are ready for a more challenging problem. Try this one.

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).

Answer on next slide...


8

October 31, 2012

May 188:24 AM

+

=

=

=

=

Here's the solution...

Is it ok if you used a

different denominator

than me?

May 188:32 AM

=

=

=

=

Try this one...Watch out for the ‐ sign!!!

May 218:07 AM

What if there are 3 fractions???

May 188:32 AM

Homework:p. 185 #4-40 eoe

Oct 179:49 AM May 162:29 PM

More Rational Expressions


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October 31, 2012

May 2110:13 AM

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.

Example #1:

Answer on next slide...

May 162:29 PM

Doesn't factor or simplify, so you are

done!!!Find your numerators by multiplying and then combine like terms.

Include all f

actors of

the denomin

ators as

your commo

n

denominator

.

May 2110:36 AM

Try one on your own...See how far you can get! Remember to include all factors of the denominator

in the new denominator .

May 2110:42 AM

Here are 3 more examples for you to attempt...

May 218:14 AM

Homework: p. 185 # 52 62 evens

Oct 179:50 AM


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October 31, 2012

May 162:29 PM

Skip this section for 2012 13

May 162:29 PM

More Rational Expressions

May 218:14 AM

Example #1:

Today we will add & subtract fractions with monomial and binomial denominators that are factorable!!!

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.

May 218:23 AM

Example #1:Factor each denominator.

Take all "pieces" of each denominator & find

new numerators.

Combine like terms and simplify your answer.

May 2110:46 AM

How about this one:

What does the bottom factor into?

What is the common

denominator?

Who are the "new"

numerators?

May 2111:43 AM

=

=


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October 31, 2012

May 211:11 PM

Uh oh...I forgot how to factor the difference of 2 perfect squares...Help me!

May 212:37 PM

Last one...Explain to me how to start this problem.

Now finish it! Tick tock, tick tock...

May 2110:14 AM

Homework: p.194 #2‐ 10 even

Quiz tomorrow!!!

May 162:30 PM

Quiz #5 today

CR #5 due tomorrow at the

beginning of class!

May 162:30 PM

Complex Fractions(Man this sounds hard!)

May 212:46 PM

You can think of complex fractions as "stacked" fractions because they are fractions stacked on top of each other.

For example, look at the following example:

This complex fraction is formed by the quotient of 2 fractions. Do you remember how to divide fractions?

Numerator

Denominator

1 1 1

2 21


12

October 31, 2012

May 228:08 AM

What if the complex fraction looks like this:

What do we do?Here's the new

complex fraction. Now divide and

simplify.

See next page...

May 2210:39 AM

1

1

Copy Change Flip.

Rewrite as 2 separate fractions.

Cancel when you can.

Write final answer.

May 228:25 AM

Example:

May 212:49 PM

Example: WHY???

May 2210:43 AM

This is another method of simplifying complex fractions.

You can still do it the same way we did before

and get the same answer.

Can you explain what this person

did?

See the next slide if you want to see how to do it the way we have been doing it...

May 237:55 AM

Example:

(y x)(y + x)

New complex fraction. Now use

your rules for dividing rational expressions.

Using the method we are used to....


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October 31, 2012

May 212:49 PM

Example:

May 212:46 PM

Homework:p. 194 #42 52 evens

May 162:30 PM

Dividing Polynomials & Synthetic Division

May 2210:52 AM

Certainly, you must remember how to divide polynomials, right???

May 2210:58 AM

Here is another example. But how on Earth do I do this one? I can't factor the numerator or the

denominator .HELP!!!

We will use a method called Synthetic Division .

Synthetic Division is a method used to divide polynomials. Sometimes the polynomials will have

common factors, and therefore divide evenly. Other times it will not divide evenly, and therefore will have a

remainder.

May 2211:29 AM

Synthetic Division:1. Start by drawing an old school long division symbol . (See below)2. Use only the coefficients of all terms in the numerator. Fill in as shown in the diagram.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 diagram).5. Begin the long division process .

x3 x2 x #Root

#


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October 31, 2012

May 2211:45 AM

x3 x2 x #Root3 -61-21

1

3

1

3

4

126

OK so here is the problem again:

Once you bring down the first

#, you start multiplying and combining, just

like in long division!

That's great, but I still don't get what these #'s mean...where is my answer?

May 221:13 PM

Remember the original question. You were dividing an x3 by an x. Doesn't that give you x2? Here's what you do...use the #'s as coefficients but start from 1 less than the original degree.

x3 x2 x #Root3 -61-21

1

3

1

3

4

126

= 1x2+ 2x + 4, R= 6

The last # is always the

remainder. If this # = 0, then the polynomials divided evenly.

May 238:14 AM

Still not quite sure about this synthetic division stuff? Let's try an easier one...

= 2x - 3

-5 2 7 -15

2 -3 0 -10 +15

Since this # is 0, that means ﴾x + 5﴿ divides evenly into

﴾2x2 + 7x 15﴿. You can prove this by factoring ﴾use arc method﴿!

May 2310:41 AM

Give this one a try on your own. Look at the steps in your notebook and follow them!!!

﴾x2 7x 78﴿ ﴾x + 6﴿

May 2210:55 AM May 2210:58 AM

1 0 0 -82

1 2 4 02 4 8

=

= x2 + 2x + 4

Since the last # is 0, the polynomials divide evenly. Check your answer by multiplying

﴾x 2﴿﴾x 2 + 2x + 4﴿.

Remember to drop 1 from the original degree of the polynomial and

use the #'s as coefficients!


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October 31, 2012

May 2210:58 AM

Give this one a try on your own. Don't forget to have a # for every "place holder."

May 2210:58 AM

Everyone can do these two...Give them a shot!

﴾2x3 5x2 4x + 6﴿ ﴾x 2﴿

﴾x4 7x 6﴿ ﴾x + 1﴿

May 2210:52 AM

Homework:p. 202 #54 64 evens

May 162:31 PM

Solving Fractional Equations

May 2310:52 AM

Remember how to solve this?

Who (or what #) is bothering you?

How can you make that # go bye bye?

May 2310:55 AM

1.Wait, this is an easy

one...Don't I just Cross Multiply???Why can I just cross multiply?


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October 31, 2012

May 232:38 PM

2.Can I just cross multiply?

May 232:38 PM

3.Can I just cross multiply?What can I do to make this

equation much easier to solve?

May 232:38 PM

4.

Whoa...this one is tricky.How many answers will I get? Why???

May 232:39 PM

5.

May 2310:55 AM

Homework:p. 209 #2- 26 eoe

May 162:31 PM

Worksheet on Fractional Equations


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October 31, 2012

May 162:31 PM

Review

Exam #3 tomorrow

May 298:32 AM

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Unit 3 notes rational expressions