Algebra (Period 1)

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 2 & 4)

Prime Factorization & GCF 4-3


Prime Factorization
Now that you know what a number divides by, 
you can get its prime factorization


Prime = a number with exactly 2 factors (itself and 1)


Composite = a number with at least 3 factors


1 and 0 are neither composite nor prime


How do you find all the prime factors of a number?

You learned Factor Trees in previous years and Factor Trees are a good strategy

Make sure you take all the bottom factors only!


EXAMPLE: 54




The prime factorization of 54 = 2x3x3x3 or 2(3³)


Make sure you list the factors from least to greatest!

There are multiple Factor Trees based on what you decide to use as your first 2 factors. But each one will get you to the same "bottom" of the tree.

The prime factorization of 54 IS ALWAYS = 2x3x3x3 or 2(3³) no matter how you start your Factor Tree

You can also use something called INVERTED DIVISION.

This is similar to Factor Trees but you must begin with the SMALLEST PRIME FACTOR that goes into the number and keep using it until it no longer works.

Then you go on to the next higher prime factor, etc.

EXAMPLE: For 54 again, but this time using Inverted Division.


THERE IS ONLY ONE CORRECT FORM OF INVERTED DIVISION!
BUT... the factors will always be in order if you follow the rules!!

Pre Algebra (Period 2 & 4)

Exponents 4-2

2⁶ represents 2⋅2⋅2⋅2⋅2⋅2 = 64

the 2 in 2⁶ is the base
the 6 in 2⁶ is the exponent
and together 2⁶ it is the power!!
The base is used as a factor 6 times to produce 64

There are 3 forms here:
2⁶ is in exponential notation
2⋅2⋅2⋅2⋅2⋅2 is in expanded notation
64 is in standard notation

You must use integer rules-- even when raising a negative number to a power!!

odd power = negative
even power = positive
(-5)³ = (-5)(-5)(-5) = -125

(-5)⁴ = (-5)(-5)(-5)(-5) = 625


an odd number of negative signs or an odd power---> it's negative
an even number of negative signs or an even power ---> it's positive

(-2)³ = ( -2)(-2)(-2) = -8
(-2)⁴ = (-2)(-2)(-2)(-2) = 16

If there is a negative BUT NO parenthesis it is ALWAYS negative

-2⁵
is read as the opposite of 2⁵

Look at what the exponent is touching.. and in this case it is just the 2!!
You can also think of -2⁵ as (-1)-2⁵

so this is -1⋅2⋅2⋅2⋅2⋅2 = -32

-2⁴ = -2⋅2⋅2⋅2 or -1⋅2⋅2⋅2⋅2 = -16
BUT REMEMBER
(-2)⁴ = (-2)(-2)(-2)(-2) = +16


THis works with variables as well. WHen you substitute in for variables put the number in parenthesis!! Hugs are important in life-- and equally important in math!!
For example Solve for x
x⁴ - 10 when x = -2
(-2)⁴ - 10
+16 - 10 = 6
But what happens when the expression is
-x⁴ - 10? Still solve for x , when x = -2
This time
the exponent is touching just the x
-(-2)⁴ - 10
You must use PEMDAS and do the exponent first!!
(-2)⁴ = 16 so substitute that back in
-(16) - 10
becomes
-16 -10 = -26

What about 4(2y -3)² when y = 5
substitute in
4[2(5) -3]²
4(10-3)²
4(7)²
4(49) =196


How about
-2x³ + 4y . When x = -2 and y = 3

-2(-2)³ + 4(3)
-2(-8) + 12
16 + 12 = 28

Algebra (Period 1)

Trinomial Squares 6-3

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

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

(a + 3)(a + 3) = a² + 3a + 3a + 9
a² + 6a + 9 (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: a²
 + 6a + 9
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. (or using the BOX method)

x² - 14x + 49
(x -7)²

16x² - 56xy + 49y²
ask yourself those important questions. They are all YES... so
(4x - 7y)²

x² -4xy + 4y²
(x - 2y)²
y⁶ + 16y³ + 64
(y³ + 8)²

9a⁸ - 30a⁴b + 25b²
(3a⁴ -5b)²

What about
2x² -40x + 200
What's the first thing you must look for? ALWAYS!!! the GCF
2(x² -20x + 100) now it is a trinomial square
2(x -10)²

similarly with
2x² -4x + 2
2(x² -2x + 1)... again NOW it is a trinomial square
2(x -1)²

18x³ + 12x² + 2x
What's the GCF? 2x

2x(9x² + 6x + 1)
2x(3x + 1)²

But look at
(a +4)² - 2(a +4) + 1
How can that be a trinomial square?
Well, think about the following
A² + 2AB + B² and
A² - 2AB + B²
represent generic trinomial squares
so if we let (a + 4) represent A... it works
[(a +4) -1]²
but we can simplify that to
(a +3)²

You could multiply everything out and then combine like terms,
that is,
(a +4)² - 2(a +4) + 1 becomes
a² -8a + 16 -2a -8 + 1 which then simplifies
a² -6a + 9 which is definitely a trinomial square
(a-3)²
but that's what we got using a simpler method!!

Try
(y+3)² + 2(y+3) + 1
It is a trinomial square in the form
A² + 2AB + B²
so [(y + 3) +1]²
which becomes (y +4)²

Math 6 Honors (Period 6 and 7)

Triangles 4-4
A triangle is the figure formed when three points, not on a line are jointed by segments.

Triangle ABC ΔABC
Each of the Points A, B, C is called a vertex
(plural: Vertices) of ΔABC

Each of the angles angle A. angle B. and angle C is called an angle of ΔABC

In any triangle-
The sum of the lengths of any two sides is greater than the length of the third side
The sum of the measures of the angles is 180

There are several ways to name triangles. One way is by angles

Acute Triangle
3 acute angles

Right Triangle
1 right angle

Obtuse Triangle
1 obtuse angle

Triangles can be classified by their sides

Scalene Triangle
no 2 sides congruent

Isosceles Triangle
at least 2 sides congruent

Equilateral Triangle
all 3 sides congruent


The longest side of a triangle is opposite the largest angle and the shortest side is opposite the smallest angle. Two angles are congruent if and only if the sides opposite them are congruent.

Algebra (Period 1)

Difference of Two Squares 6-2

Again, remember that FACTORING just UNDOES multiplication.

In this case, the multiplication that you'll be UNDOING is FOILING.

FOIL:
(a + b)(a - b)

You will get:
a² - b²
This is the DIFFERENCE (subtraction) of TWO SQUARES.

Now FACTOR:
a² - b²
You undo the FOILING and get:
(a + b)(a - b)


REMEMBER:
You must have two different signs because that's how the MIDDLE TERM disappears!

You will get ADDITIVE INVERSES which will become ZERO



HOW TO RECOGNIZE THE DIFFERENCE OF TWO SQUARES:

1) Is it a binomial?

2) Is it a difference?

3) Are both terms perfect squares?


IF YES TO ALL 3 QUESTIONS, THEN YOU HAVE A DIFFERENCE OF 2 SQUARES!!

HOW TO FACTOR THE DIFFERENCE OF 2 SQUARES:

1) Double hug  (    )(    )

2) Find square root of each term (sq rt sqrt)(sq rt sq rt)

3) Make one sign positive and one sign negative.
              
(sq rt + sqrt)(sq rt -sq rt)


                           
Of course, they get more complicated! 
We can combine pulling out the GCF with this!

ALWAYS LOOK FOR A GCF TO PULL OUT FIRST!!!!!!

EXAMPLE:

27y² - 48y⁴ 

First, look for a GCF that can be pulled out.

The GCF = 3y²

Factor out the GCF (look at Chapter 6-1):
3y²(9 - 16y² )
NOW YOU HAVE A DIFFERENCE OF TWO SQUARES TO FACTOR:


3y²(3 - 4y)(3 + 4y)


CALLED FACTORING COMPLETELY BECAUSE
 YOU CANNOT FACTOR FURTHER!


Always check your factoring by distributing or FOILing back!


THERE IS NO SUCH THING AS THE SUM OF TWO SQUARES!

a² + b² CANNOT BE FACTORED!!!!!


BUT - b² + a²
= + a² -b²
= (a + b)(a - b)
BECAUSE IT'S JUST SWITCHED (COMMUTATIVE)

Algebra (Period 1)

Factoring Polynomials 6-1


Chapter 5 was a very important building block of Algebra but

CHAPTER 6 IS EVEN MORE IMPORTANT FOR HIGH SCHOOL!!!



REMEMBER THIS KEY CONCEPT:
FACTORING WILL NEVER CHANGE THE ORIGINAL VALUE OF THE POLYNOMIAL …SO YOU SHOULD ALWAYS CHECK BY MULTIPLYING BACK!!!!
(You'll either distribute or FOIL.)


Factoring is a skill that you must understand to be successful in higher level math!!!

We did a simple version of this back in Chapter 1 and you had a RACE on it!


Factoring is simply UNDOING multiplying


Say you multiplied 5 by 10 and got 50

How would you undo it?
DIVIDE by 5!


So FACTORING uses the concept of DIVIDING.

You're actually undoing the DISTRIBUTIVE PROPERTY.

How?
You look for the most of every common factor....the GCF!

Then you pull out the GCF (divide it out of) from each term,
placing the GCF in front of ( )


EXAMPLE:

FIRST, DISTRIBUTE:
2m²n (2n2 + n + 3)
=
4m²n³ + 2m²n² + 6m²n


Now, pretend you don't want the 2m²n to be distributed anymore...

What should you end up with once you UNDO the Distributive Property?

2m²n (2n² + n + 3)


That's exactly what you started with!

So is it that easy?

Well yes... and no...

Yes because that is the answer
…and
…
No because it was only that easy because I gave you how it started!

You won't know how it started in a real problem!



THIS IS AN EXAMPLE OF THE WAY YOU WOULD USUALLY SEE IT.

The question would say:
FACTOR:
 4m²n³ + 2m²n² + 6m²n


Step 1: What does each term have in common (what is the GCF) ?

They each can be divided by 2m²n


Step 2: Put the GCF in front of a set of ( ) and divide each term by the GCF
2m²n ( 4m²n/2m²n + 2m²n²/2m²n + 6m²n/2m²n)


Step 3: SIMPLIFY and you'll get:

2m²n (2n² + n + 3)



Step 4: Check your answer!!!!!

Always check your factoring of the GCF by distributing back!


(incognito, it should be the same thing)

Another check is to make sure you have factored out the entire GCF.

Look inside the parentheses and ask yourself if there is still any factors in common between the terms.
If there is, then you haven't factored out the GREATEST Common Factor.
For example, let's say in the prior example that you only factored out 2mn instead of 2m²n. You would have:
2mn ( 4m²n³/2mn + 2m²n²/2mn + 6m²n/2mn)


= 2mn(2mn²+ mn + 3m)


If you just check by distributing back, the problem will check.

BUT ...
Look inside the ( ) and notice that each term still has a common factor of m!
So this would not be the fully factored form!
So make sure you always look inside the ( )!!!

RELATIVELY PRIME TERMS -

TERMS WITH NO COMMON FACTORS

THAT MEANS THAT THEY CANNOT BE FACTORED
(GCF = 1)

We say they are "not factorable"