Math 6H ( Periods 3, 6, & 7)

Subtracting Integers 11-3

The life story about someone who was so negative-- you wanted to take a little negativity away but since you can't do that you add a little positiveness-- works in math as well!!
Instead of subtracting ... "ADD THE OPPOSITE!"
We proved it in class with our little red and yellow tiles... If you need to review, make your own out of red and yellow paper-- or whatever colors you want!!

In life-- to take away a little negative-- add some positive

To take away an integer... add its opposite.

Rule from our textbook
for all integers a and b

a - b = a + (the opposite of b) or
a - b = a + (-b)

Instead of subtracting.. "ADD THE OPPOSITE"
make sure you do the check, check.. you need to have two check marks.. one changing the subtraction to addition and the other changing the sign of the 2nd number to its opposite.

5 - ^- 2=
5 + ⁺ 2= 7

-2 - 5 =
- 2 + ^- 5 =
before I give you the answer look... we are looking at
-2 + ^-5
We are adding two negatives.. so we are back to the rules from Section 11-2...
when adding the same sign just add the number and use their sign so
- 2 + ^- 5 = -7

But what about -2 - ^-5 ?
adding the opposite, we get
-2 + ⁺ 5 .
Now, the signs are different so the rule from Section 11-2 is
ask yourself... "Who wins?" and "By how much?"

Okay, here the positive wins so I know the answer will be +
and by how much means.. to take the difference
5-2 = 3
so -2 + ⁺ 5 = +3
Do you need to put the + sign? No, but I like to in the beginning to show that I checked WHO WON!!

What about
-120 - ^-48?
add the opposite
-120 + ⁺ 48 follows the
Different signs rule... so
ask yourself
Who wins? answer: the negative.. so I know the answer will be negative..
And "By How Much?" take the difference 120-48 = 72
so
-120 + ⁺ 48 = -72

NEVER EVER CHANGE THE FIRST NUMBER'S SIGN!!

WALK THE LINE, the number line-- that is!!
Remember... Attitude is such a little thing... but it makes a BIG difference!!
Always start with a positive attitude!!
When you walk the line, Which way are you always facing when you start???

Always attempt to get everything into addition so we can follow the rules of Section 11-2 Adding Integers.
1) SAME SIGN rule---> just add the numbers and use their sign
-4 + -5 = -9
2) DIFFERENT SIGNS rule
ask yourself those 2 important questions
a) Who wins? (answer is either negative or positive)
b) By how much? (Take the difference)

Page 376 answers to # 2-18 (evens)
2. -9
4. -5
6. 22
8. -9
10. -74
12. -32
14. -160
16. 498
18. -284

Algebra Period 4

Factoring ax² +bx + c Section: 6-5

Checklist of how to factor thus far:

1. Look for a GCF of all terms

2. Binomials - look for difference of two squares
 both perfect squares - double hug - one pos, one neg - square roots of both terms

3. Trinomials - look for Trinomial Square (factors as a binomial squared)

first and last must be perfect squares - middle must be double the product of the two square roots

SINGLE hug - square roots of both terms - sign is middle sign

4. Trinomials - last sign positive - double hug with same sign as middle term - factors that multiply to last and add to middle

5. Trinomials - last sign negative - double hug with different signs,
putting middle sign in first hug - factors that multiply to last and subtract to middle - middle sign will always be with the bigger factor

6. 4 terms - Factor by grouping - make sure in descending order - pair off the first 2 terms and the last 2 terms -
make sure there's a PLUS sign in between the 2 pairs -
factor out the GCF of each pair - if the binomial left in the (  ) is the same, it's factorable.


REMEMBER:
FACTORING WILL NEVER CHANGE THE ORIGINAL VALUE OF THE POLYNOMIAL SO YOU SHOULD ALWAYS CHECK BY MULTIPLYING BACK!!!!

Factoring ax²+ bx + c  Sections 6-5 or
FACTORING TRINOMIALS WITH A COEFFICIENT ON THE 1ST TERM.  
We'll use FACTORING BY GROUPING (Chapter 6-6)

I call this "Xbox 360" and you'll see why in class!
When you have a trinomial with a coefficient on the first term, factoring becomes more difficult.

A good method to factor is to use factoring by grouping.

1) Multiply the first term's coefficient by the last term's coefficient

2) By guess and check, find 2 factors that multiply to the product you got in #1 and add to your middle term
(just like how you did it for trinomials without a coefficient on the first term)
I like to set up a T Chart and go from there!!

3) Rewrite the original trinomial as a 4 term polynomial using the first term, the two factors you found, and the the last term.
IT SHOULD SIMPLIFY TO THE ORIGINAL PROBLEM!

4) Factor by grouping

AGAIN, NOT ALL TRINOMIALS ARE FACTORABLE!


EXAMPLE: 15n² - 19n - 10

1) Multiply 15 by 10 = 150

2) Find 2 factors that multiply to 150 and add to -19 (which means subtract to 19)

Think: 15 and 10? NO

30 and 5? NO

25 and 6? YES! 

Since the negatives must win, it must be -25 and +6 = -19!


3) Rewrite 15n² - 19n - 10 as a 4 term polynomial:
15n² - 25n + 6n - 10


4) Factor by grouping: (15n² - 25n) + (6n - 10)

5) Pull out the GCF: 5n(3n - 5) + 2(3n - 5)

(3n - 5)(5n + 2)


6) Check to make sure you can't factor any more.

7) FOIL to make sure you did not make a silly mistake!

Experimenting with Zoho

Experimenting with Zoho writer...
MAKE SURE IT'S IN DESCENDING ORDER FIRST!!!!

EXAMPLE: 6x³ + 9x² + 4x - 6

NOTICE THAT THERE IS NO GCF OF ALL 4 TERMS!

Factoring by grouping says if there is no GCF of all 4 terms, look and see if there is a GCF of just 2 terms at a time.
Put ( ) around the first 2 terms and the 2nd 2 terms:
(6x³ + 9x² ) + ( 4x - 6)
FACTOR OUT THE GCF FROM EACH SET OF TWO TERMS:

3x² (2x - 3) + 2(2x - 3)

Math 6H ( Periods 3, 6, & 7)

Adding Integers 11-2

Rules: The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.

So- if the two numbers have the same sign, use their sign and just add the numbers.

-15 + -13 = - 28

-10 + -4 = -14

Rules: The sum of a positive integer and a negative integer is :

POSITIVE… IF the positive number has a greater absolute value

NEGATIVE… IF the negative number has a greater absolute value

ZERO… IF both numbers have the same absolute value

Think of a game between two teams- The POSITIVE TEAM and The NEGATIVE TEAM.

30 + -16 … ask yourself the all important question…
“WHO WINS?
in this case the positive and then ask
“BY HOW MUCH?”
take the difference 14

14 + - 52…
“WHO WINS?”
the negative… “BY HOW MUCH?”
38
so the answer is -38


(-2 + 3) + - 6 you can work this 2 ways

(-2 + 3) + - 6 = 1 + -6 = -5 or
using all the properties that work for whole numbers
Commutative and Associative properties of addition
can change expression to (-2 + -6) + 3 or -8 + 3 = -5 you still arrive at the same solution.

You want to use these properties when you are adding more than 2 integers.
First look for zero pairs—you can cross them out right away!!
3 + (-3) = 0
-9 + 9 = 0

Then you can use C(+) to move the integers around to make it easier to add them together rather than adding them in the original order. In addition, you can use A(+) to group your positive and negative numbers in ways that make it easier to add as well.

One surefire way is to add all the positives up… and then add all the negatives up.
At this point ask yourself that all important question… WHO WINS? …
use the winner’s sign..
and then ask yourself..
BY HOW MUCH?

example:

-4 + 27 +(-6) + 5 + (-4) + (6) + (-27) + 13

Taking a good scan of the numbers, do you see any zero pairs?
YES—so cross them out and you are left with
-4 + 5 + (-4) + 13
add your positives 5 + 13 = 18
add your negatives and use their sign – 4 + -4 = -8

Okay, Who wins? the positive
By how much? 10
so
-4 + 27 +(-6) + 5 + (-4) + (6) + (-27) + 13 = 10

Pre Algebra Period 1

Comparing and Ordering Fractions 

5-1 (cont'd)

Ordering or comparing fractions:(last method)

4: Make them into decimals because DECIMALS = FRACTION posers!

(or decimals are just fraction wannabes!)
Today, we will change fractions to decimals and decimals to fractions.

How to change a fraction to a decimal
1.
Divide (ALWAYS WORKS!)

EXAMPLE: 3/4 = 3 divided by 4 = .75

If the quotient starts repeating, then put a bar over the number(s) that repeat. OR

2. Use equivalent fractions (SOMETIMES WORKS!)

Works if the denominator can be easily made into a power of 10

SAME EXAMPLE: but this time you will multiply by 25/25 to get 75/100 = .75



3. MEMORY! Some equivalencies you should just know!
EXAMPLE: 1/2 = .5


IF IT'S A MIXED NUMBER, JUST ADD THE WHOLE NUMBER AT THE END!

EXAMPLE: 8 3/4

For the fraction: 3 divided by 4 = .75

Add the whole number: 
8.75


IF THE MIXED NUMBER OR FRACTION IS NEGATIVE, SO IS THE DECIMAL!


CHANGING TERMINATING DECIMALS TO FRACTIONS:
EASY!!!
Read it, Write it, Simplify!



EXAMPLE:

Change .24 to a fraction

1) READ IT: 24 hundredths

2) WRITE IT: 24/100

3) SIMPLIFY: 24/100 = 6/25



EXAMPLE with whole number:

Change 7.24 to a fraction

The 7 is the whole number in the mixed number so you just put the 7 at the end

1) READ IT: 24 hundredths

2) WRITE IT: 24/100

3) SIMPLIFY: 24/100 = 6/25

4) 7 6/25



HOW TO CHANGE REPEATING DECIMALS TO FRACTIONS---> we will do that in 2010!!

Math 6H ( Periods 3, 6, & 7)

Negative Numbers 11-1

On a horizontal number line we use negative numbers for the coordinates of points to the left of zero. We denote the number called ‘negative four’ by the symbol -4. The symbol -4 is normally read ‘ negative 4’ but we can also say ‘ the opposite of 4.’

The graphs of 4 and -4 are the same distance from 0—>but in opposite directions. Thus they are opposites. -4 is the opposite of 4.

The opposite of 0 is 0

Absolute Value is a distance concept. Absolute value is the distance of a number from 0 on a number line. The absolute value of a number can NEVER be negative!!

Counting (also known as Natural) numbers: 1, 2, 3, 4, ….
Whole numbers 0, 1, 2, 3, 4….
Integers are natural numbers and their opposites AND zero
…-4, -3, -2, -1, 0, 1, 2, 3, 4….

The opposite of 0 is 0.

The integer 0 is neither positive nor negative.

The farther we go to the right on a number line--- the bigger the number. We can compare two integers by looking at their position on a number line.

if x < 0 what do we know? x is negative number
if x > 0, what do we know? x is a positive number

We have been practicing representing integers by their graphs, that is, by points on a number line.

Make sure that your number line includes arrows at both ends and a line indicating where zero falls on your number line.
The graph of a number MUST have a closed dot right on the number line at that specific number.
Please see our textbook page 366 for an accurate example.

Pre Algebra Period 1

Comparing and Ordering Fractions

 5-1
HOW TO FIND THE LEAST COMMON MULTIPLE:

LCM = Smallest number that your numbers can go into.
Just like GCF, let's look at the letters backwards to understand it!

Multiple = each number given in the problem must go into this number (example for multiples of 2: 2, 4, 6, 8.... multiples of 3: 3, 6, 9, 12...     multiples of 5: 5, 10, 15, 20...)

Common = must be a number that ALL THE NUMBERS go into 

Least = must be the SMALLEST number that ALL THE NUMBERS go into

There are the same ways to find it as the GCF:

1) List multiples of each number and circle the smallest one that is common to all the numbers
---> most of the time takes WAY TOO long!!
2) Circle every factor in the prime factorizations of each number that is different and multiply

3) List the EXPANDED FORM prime factorizations in a table and bring down ONE OF EACH COLUMN. Then multiply. (or you can do this with exponential form but you need to bring down the HIGHEST POWER of each column).
4) Using the BOX method from class create an L from the left side and the bottom row of relatively prime factors. It is their product.

THE DIFFERENCE BETWEEN GCF AND LCM:
For the GCF, you need the LEAST POWER of only the COMMON FACTORS.
For the LCM, you need the GREATEST POWER of EVERY FACTOR.

WHY DO WE NEED EVERY FACTOR THIS TIME?

Because it's a multiple of all your numbers!

Multiples start with each number, so all the factors that make up each number have to be in this common multiple of all the numbers.

For example, say we're finding the LCM of 12 and 15, that multiple must be a multiple of 12: 12, 24, etc.
AND 15: 15, 30, etc.
So the COMMON multiple must include 12 (2x2x3) AND 15 (3x5).
The LCM must have two 2s and one 3 or 12 won't go into it.
It must also have that same 3 that 12 needs and one 5 or it won't be a multiple of 15.
EXAMPLE: 54 and 36

LIST MULTIPLES OF EACH NUMBER UNTIL YOU SEE ONE THAT BOTH OF THEM GO INTO (the LEAST MULTIPLE that is COMMON to both)
36 = 36, 72, 108, 144 ...
54 = 54, 108
This more difficult than the listing method for GCF for 2 reasons: You don't know where to stop as you least the first number...multiples go on forever! and the numbers get big very fast because they're MULTIPLES, not factors!
PRIME FACTORIZATION METHOD
54 = 2x3x3x3

36 = 2x2x3x3
The LCM will be one of each factor of each number (but don't double count a factor that is common to both numbers...once you have it in your LCM, check it off in the other number if it's common...you don't need that factor again)

LCM = 2x2x3x3x3 = 108
Let's look at this calculation and think about why it works:
Why does it need TWO 2s?
Although 54 only needs ONE, 36 needs TWO or 36 won't go into the LCM.
Try it putting in only ONE 2: 2x3x3x3 = 54. Does 36 go into 54? NO
Why does the LCM need THREE 3s? Although 36 needs only TWO 3s, 54 needs THREE. Try it putting in only TWO: 2x2x3x3 = 36.
36 is TOO SMALL to even be a MULTIPLE of 54!
WHAT IF BY MISTAKE YOU "DOUBLE" UP FACTORS AND USE ALL OF THEM?
For the GCF, you would have got a number way too big to be a factor of the numbers. For the LCM, you will STILL GET A COMMON MULTIPLE! But it WON'T BE THE LEAST!
In fact if you double up, generally you'll get a really big number and the bigger the number is, the harder it is to use.
For example, if you use all the factors of both 36 and 54, you're just multiplying 36 x 54 = ???? 
1944!!!! 
Would you rather use 108 or 1944???


LCM WITH A COLUMN APPROACH:
You place the prime factorization for each number in columns like we did for the GCF, matching the factors in each column.
If a factor doesn't match, it gets a separate column.
YOU'LL JUST TAKE ONE OF EACH COLUMN AND MULTIPLY!

                                    SAME EXAMPLE: 54, 36
                                     
36 = 2 x 2 x 3 x 3
                                     
54 = 2 x       3 x 3 x 3
                                   
LCM = 2 x 2 x 3 x 3 x 3
= 108
You can do it in exponential form, too.

If you do it in exponential form, you take the HIGHEST POWER of each column!

Works with variables the exact same way!
Take the HIGHEST power of EVERY variable (not just the ones that are common like the GCF)
YOU CAN FIND BOTH THE GCF AND THE LCM IN THIS SAME COLUMN FORMAT!

CHECKING THE LCM TO MAKE SURE IT WORKS:

Algebra Period 4

Factoring by Group 6-6

First a review:
Checklist of how to factor thus far-->

1. Look for a GCF of all terms

2. Binomials - look for difference of two squares

both perfect squares - double hug - one positive, one negative - square roots of both terms

3. Trinomials - look for Trinomial Square (factors as a binomial squared)

first and last must be perfect squares - middle must be double the product of the two square roots

SINGLE hug - square roots of both terms - sign is middle sign

4. Trinomials - last sign positive - double hug with same sign as middle term - factors that multiply to last and add to middle

5. Trinomials - last sign negative - double hug with different signs, putting middle sign in first hug - factors that multiply to last and subtract to middle - middle sign will always be with the bigger factor



REMEMBER: 
FACTORING WILL NEVER CHANGE THE ORIGINAL VALUE OF THE POLYNOMIAL SO YOU SHOULD ALWAYS CHECK BY MULTIPLYING BACK!!!!

(we're skipping 6-5 and then going back to it)
When you have 4 TERMS IN YOUR POLYNOMIAL!


You put the polynomial in 2 sets of 2 by using ( )

Then you factor out the GCF for each set of 2 terms individually


DOES THIS ALWAYS WORK FOR EVERY 4 TERM POLYNOMIAL?

Of course not!
But for this section of the math book, it will!

What happens if it doesn't work? The polynomial may just not be factorable!

MAKE SURE IT'S IN DESCENDING ORDER FIRST!!!!

EXAMPLE: 6x³ - 9x² + 4x - 6
First notice there is NO GCF of all the terms!!

Factoring by grouping says if there is no GCF of the 4 terms, look and see if there is a GCF of just 2 terms at a time!!

Put ( ) around the first 2 terms and another ( ) around the 2nd set of terms.
(6x³ - 9x²) + (4x - 6)
Factor out the GCF from each set of two terms
3x²(2x - 3) + 2(2x - 3)
Look for a COMMON factor to factor out between the two sets
In this case its (2x - 3)
Pull out
(2x - 3)(3x + 2)
and check to make sure you cannot continue to factor!!

Math 6H ( Periods 3, 6, & 7)

Least Common Multiple 5-6

Also check out December 4, 2009 posting of LCM!! Here is a review of that lesson...
Let’s look at the nonzero multiples of 8 and 12—listed in order
Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72…

Multiples of 12: 12, 24, 36, 48, 60, 72, ….

The numbers 24, 48, and 72, ... are called common multiples of 8 and 12. The least of these multiples is 24 and is therefore called the least common multiple.

LCM(8, 12) = 24

To find the LCM of two whole numbers, we can write out lists of multiples of the two numbers.

Or, we can use prime factorization

Lets find LCM(12, 15)

12 = 2²∙3
15 = 3∙5

The LCM will be made up of the greatest power of each factor

LCM will be 2²∙3∙5 = 60
The book has a third option or method
you can check out, if you’d like

Let’s find LCM (54, 60)


54= 2∙3∙3∙3 = 2∙3³
60 = 2∙2∙3∙5 = 2²∙3∙5

The greatest power of 2 that occurs in either prime factorization is 2²
The greatest power of 3 that occurs in either prime factorization is 3³
The greatest power of 5 that occurs in either prime factorization is 5
Therefore, LCM(54,60) is 2²∙3³∙5 = 540


REMEMBER:
The GCF (greatest common factor) is a factor. The GCF of two numbers will be either the smaller of the two or smaller than both

The LCM (least common multiple) is a multiple. The LCM of the two numbers will be the largest of the two or larger than both.

Algebra Period 4

Factoring x² +bx + c or Factoring Trinomials 6-4

You are reversing it back to BEFORE it was FOILed.
Always check your factoring by FOILing or BOXing back!!


Factoring Trinomials with a:
PLUS sign as the second sign
x² + bx + c
Following these steps:

1. set up your hugs ( )( )
2. When the last sign is positive the BOTH signs in each of the ( )( ) are the SAME!!
3. How do you know what those two signs are? It is whatever the sign is of the 2nd term of the trinomial. Put that sign in BOTH parentheses.
4. to factor (unFOIL), you will need to find two factors that
MULTIPLY to the LAST term and
ADD to the MIDDLE term

you can set up a box with

___ X ___ =
___ + ___ =

and fill in the blanks.
I suggest you make a T-chart with all the factors of the last term-- using your divisibility rules!!
Example:
x² +8x + 15
Follow the steps
( )( )
Think: last term sign is + so both signs are the same
Think: first sign (sign of the 2nd term) is + so both signs are positive
put + into the ( )( )
( + )( + )
You already know the "F" in FOIL means that both first terms must be x --> so put those terms in
(x + )(x + )
Now to get to the L in FOIL you need two factors whose product is 15. This is easy but using a T chart
15
1 I 15
3 I 5

you see 1 X 15 or 3 X 5 are possibilities

BUT, you also need two numbers to add to the I and O of FOIL which means that the two numbers must add up to 8 ( the middle term)
Since 3 + 5 = 8, they must be the two factors that will work
3 X 5 = 15
3 + 5 = 8
(x + 5) (x +3)

At this point it does not matter which factor you put into the first ( ) because they are the SAME sign but I always tend to put the LARGER number in the first ( ) because of other rules -- which you will learn later this week)

Next example
x² - 8x + 15
Follow the steps
( ) ( )
THINK: Last sign is + so the signs are the same
THINK: First sign ( 2nd term) is NEGATIVE so BOTH signs are NEGATIVE
( - ) ( - )

Again,
You already know the "F" in FOIL means that both first terms must be x --> so put those terms in
(x - )(x - )

Now to get the "L" in FOIL, you need two factors whose product is 15
BUT, you also need two numbers to add to the I and O of FOIL which means that the two numbers must add up to -8 ( the middle term)
Since 3 + 5 = 8, they must be the two factors that will work
-3 X -5 = 15
-3 + -5 = -8
(x - 5)(x - 3)
(It doesn't matter which is first because they're the same sign!)
Now FOIL to see if we're right!

Last example:
x² - 8xy + 15y²
Same problem as the one above except now there are two variables. Simply use the same steps above and include the y

(x -5y)(x -3y)


Factoring Trinomials with a:
NEGATIVE sign as the second sign

x² + bx - c
We will use the same method as yesterday to factor these basic trinomials!
1.
Set up your (       )(       )

2. Look at the SECOND or last sign

If it's negative, then the signs in the (   ) are DIFFERENT
Why?
Because when you multiply integers and get a NEGATIVE product, the only way that will happen is if they are DIFFERENT signs.
Remember that the last term is the product of the two LAST terms in FOILing.


3. Now look at the sign of the second term.

It tells you "Who wins," meaning which sign must have the larger absolute value. 
Remember that the middle term is the SUM of the "O" and the "I" terms when FOILing.
Because these two terms have DIFFERENT signs, when you add them, you actually "subtract" and take the sign of the larger absolute value.
( This is just integer rules!!)
Put that sign in the first parentheses and always put the bigger number in the first parentheses.
4. To UNFOIL (factor), you will need to find 2 FACTORS that MULTIPLY to the last term, but SUBTRACT to the middle term.
(yesterday the factors needed to ADD to the middle)

Or you can still say you're adding, but since they are DIFFERENT signs, you will end up subtracting!

This is still an educated guess and check!

To help you do this, I suggest to set it up like this:

____ x ____ = ____
____
 - ____ = ____

Again, setting up a T-chart with all the factors also helps you visualize the two numbers you are looking for!!



EXAMPLE:

x² + 2x - 15
( ) ( )
THINK: LAST sign is - so the signs are DIFFERENT
THINK: First sign is + so the POSITIVE WINS!!
( + ) ( - )
You know the "F" in FOIL means that both the fist terms must be x so
(x + )(x - )
Now to get the "L" in FOIL, you need 2 factors whose product is NEGATIVE 15
(Don't forget the sign1!!)
Several possibilities like 1 X -15 or 15 X -1 or 3 X -5 or 5 X -3
BUT since the POSITIVE must win , according to the middle term of the example (+2x)
you know that the bigger factor must be positive ( so it can win!!)
Therefore your choices are POSITIVE 15 X NEGATIVE 1 or POSITIVE 5 X NEGATIVE 3
BUT, you also need them to ADD to the I and O in FOIL so pick the two factors that also ADD to POSITIVE 2
Since -3 + 5 = +2 these must be the two factors that will work
I set it up like this:

____ x ____ = 15

____ - ____ = 2

so
5 x 3 = 15

5 - 3 = 2

YOU CAN ALSO DO THIS WITH THE APPROPRIATE SIGNS and adding:

+____ x -____ = -15

+____ + -____ = + 2

so

5 x (-3) = -15

5 +( -3) = 2


THIS IS WHERE IT DOES MATTER WHICH NUMBER YOU DO HAVE WITH THE SIGN BECASUE THE + MUST WIN!!
( x + 5 )( x - 3 )



NEXT EXAMPLE:

x² - 2x -15

(      )(      )

THINK: Last sign is - so signs are DIFFERENT!

THINK: First sign is - so NEGATIVE MUST WIN

(    -   )(    +   )

You know the the "F" in FOIL means that both first terms must be x

( x - )( x + )

Now to get the "L" in FOIL, you need 2 factors whose product is NEGATIVE 15

Like 1 and 15, or 3 and 5

But you also need to add to the I and O in FOIL which means that the two factors
 must add to NEGATIVE 2

Since 3 + -5 = -2, this must be the two factors that will work:

( x - 5 )( x + 3)

It matters which number you have with which sign because the negatives must win!

That's why I always put the sign of the middle term in the first parentheses.

That way, I always know to put the larger number in the first parentheses, so that sign will win.

Now FOIL to see if we're right!


LAST EXAMPLE:

x² - 2xy -15y²
Same problem as the one before, except now there are 2 variables!

Simply use the same factorization and include the y

( x - 5y )( x + 3y)


ALWAYS CHECK BY FOILing or BOXing Back!!

Pre Algebra Period 1

Rational Numbers 4-6

THE NUMBER SYSTEMS

Natural (counting): 1, 2, 3, ....

Whole: 0, 1, 2, 3, ...

Integers: ...-3, -2, -1, 0, 1, 2, 3, ...

Rational: All numbers that can be written as a fraction or RATIO (decimals-terminating and repeating, fractions, mixed numbers, integers)

Irrational: Decimals that never terminate or repeat - like pi and square root of 2


Real numbers: All the number systems together



Complex or Imaginary Numbers: we don't study these until Algebra II, but they are stated as

a + bi where i is equal to the square root of -1



Again, Rational number = any number that can be expressed as the ratio (fraction) of two integers

a/b, where a and b are both integers and b cannot be zero

b cannot be zero because you cannot divide by zero.....IT'S UNDEFINED!


There are positive and negative rational numbers.

Algebra Period 4

Trinomial Squares 6-3

This is a special product that we learned in Chapter 5 when we did FOILing.

FOIL:
(3 + a)² (called a binomial squared)

(3 + a)(3 + a) = 9 + 3a + 3a + a²
=
9 + 6a + a² (called a trinomial square)



Again, you see that the middle term is DOUBLE the product of the two terms in the binomial, and the first and last terms are simply the squares of each term in the binomial.
   
HOW TO RECOGNIZE THAT IT IS A BINOMIAL SQUARED:

1) Is it a trinomial? (if it's a binomial, it cannot be a binomial squared - it may be diff of 2 squares)

2) Are the first and last terms POSITIVE?
3) Are the first and last terms perfect squares?

4) Is the middle term double the product of the square roots of the first and last terms?


IF YES TO ALL OF THESE QUESTIONS, THEN YOU HAVE A TRINOMIAL SQUARE
 
TO FACTOR A TRINOMIAL SQUARE: 9 + 6a + a²

1) (     )²

2) Put the sign of the middle term in the (   +   )²

3) Find the square root of the first term and the last term and place in the parentheses:   (3 + a)²

4) Check by FOILing back.

Math 6H ( Periods 3, 6, & 7)

Greatest Common Factor 5-5

Also check out December 1, 2009 posting of GCF. Here is a review of that lesson...


When the factors in the numbers 30 and 42 are listed, the numbers 1, 2, 3, and 6 appear in both lists
Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42

These numbers are called common factors of 30 and 42. The number 6 is the greatest of these numbers and is therefore called the greatest common factor of the two numbers.
We write
GCF(30,42) = 6
to denote the greatest common factor of 30 and 42

Find GCF(54, 72)
List the factors of each number

54: 1, 2, 3, 6, 9, 18, 27, 54
Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72

the common factors are 1, 2, 3, 6, 9, and 18

the greatest number in both lists is 18. Therefore,

GCF(54, 72) = 18

Another way to find the GCF of two numbers is to use prime factorization.
Find GCF (54, 72)
First find the prime factorization of 54 and of 75

54 = 2∙3∙3∙3∙3
72 = 2∙2∙2∙3∙3

Find the greatest power of 2 that occurs in both prime factorization. 2
Find the greatest power of 3 that occurs in both prime factorization 3²
Therefore GCF(54,72) = 2∙3² = 18

Try GCF(45, 60) using the prime factorization method

The number 1 is a common factor of any two whole numbers.
If 1 is the GCF, then the two numbers are said to be relatively prime.
Show that 15 and 16 are relatively prime
List the factors of each number
Factors of 15 = 1, 3, 5, 15
Factors of 16: 1, 2, 4, 8, 16
Since the GCF(15, 16) = 1, the two numbers are relatively prime

Pre Algebra Period 1

Simplifying Fractions 4-4

Equivalent Fractions - Just multiply the numerator and the denominator by the same number and you will get an equivalent (equal) fraction to the one you started with.


GOLDEN RULE OF FRACTIONS = Do unto the numerator as you do unto the denominator


Simplifying fractions (your parents call this "reducing")

2 good ways:

(1) Just divide both the numerator and denominator by the GCF

(2) Another way: Rewrite the numerator and denominator in prime factorization form. Then simply cross out each common factor on the top and bottom
(they cross out because it's 1)

You'll be left with the simplified fraction every time!!!!
                                           


THE GCF METHOD:

One of the reasons we learn the GCF is because it's the FASTEST WAY TO SIMPLIFY FRACTIONS IN ONE STEP!!!

Just divide both the numerator and denominator by the GCF

THE PROBLEM WITH THE METHOD:
If you're not comfortable finding the GCF, you're pretty much sunk with this method! :(

THE BEST REASON TO USE THIS METHOD (other than it's a Calif. STAR Key Standard), it truly is the FASTEST :)

So imagine you have a "GCF Magical Voice" in your head...
The voice tells you the GCF of the numerator and the denominator...
You simply use that GCF to divide both the top and bottom of your fraction and you're done in one step!



THE PRIME FACTORIZATION METHOD:

This is sort of using the GCF "incognito" (in disguise)!

Rewrite the numerator and denominator in prime factorization form.

(Use a Factor Tree or Inverted Division to find the Prime Factorization if necessary).

Then simply cross out each common factor on the top and bottom.

(You're actually using the ID Property of Multiplication because 
each "crossout" is really a quotient of 1!)

You'll be left with the simplified fraction every time!!!!

If you actually multiplied together all your cross-outs, you'd get the GCF...
so you're using the GCF without even computing it!

THE PROBLEM WITH THIS METHOD:
You may think it's a lot of work


THE BEST REASON TO USE THIS METHOD: Although it takes time, everyone can do a Factor Tree or Inverted Division and create the Prime Factorization...
You'll never get the wrong answer with this one!



THE CROSS OUT METHOD:

You simply think of the first number that comes to your mind that "goz-into" both the numerator and the denominator and keep going until it's simplified.

If it's even, most people start with dividing it in half....and then in half again, etc.

This probably takes the longest, but in practice, most people use this method!

THE PROBLEM WITH THIS METHOD: You may think that a fraction is simplified, but you've missed a factor...this especially happens when the number is odd and you're always used to using 2 to divide the top and the bottom!

THE BEST REASON TO USE THIS METHOD: No one ever forgets how to do this method...it just comes naturally and there are no "precise" steps to do!
 
EXAMPLE: Simplify by each method:
36/
54
 
GCF METHOD:

The GCF is 18:

36 ÷ 18 = 2

54 ÷ 18 = 3



PRIME FACTORIZATION METHOD:

36 = 2 x 2 x 3 x 3

54 = 2 x 3 x 3 x 3

Two of the 3s cross out and one of the 2s

You are now left with:

2/
3

That's it!!!!!!!!!



CROSS OUT METHOD:
36 ÷ 2 = 18 ÷ 3 = 6 ÷ 3 = 2

54 ÷ 2 = 27 ÷ 3 = 9 ÷ 3 = 3


so 36/54 = 2/3
Do the same thing with variables!