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Our Class Blog

Welcome to our class blog... where you can earn extra credit by adding your own relevant comments about our class notes for the day... or where you can find answers from others in your class. Check here often, especially if you have been absent. You might just find out the math strategy that works for you!!

Algebra Honors

Factoring the Difference of Two Squares  8-8

Again… remember that FACTORING just UNDOES multiplication

Chapter 8-3 Déjà vu  Foil the following:
(a + b)(a –b) you will get a² – b²

This is the DIFFERENCE 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 DIFFERENCE of TWO SQUARES

1) Is it a binomial?
2) Is it a difference?
3) Are both terms perfect squares?

If YES to all three questionsàthen you have DIFFERENCE of TWO SQUARES

How to factor the DIFFERENCE of TWO SQUARES

1) Put a set of double {{HUGS}}  (      )(       )
2) Find the square root of each term  (SQRT   SQRT) (SQRT   SQRT)
3) Make one sign positive and one sign negative  (SQRT+SQRT) (SQRT-SQRT)

Of course, they can get more complicated
Always look for a GCF FIRST to pull out

EXAMPLE:

27y² – 48y⁴

At first this just looks like a binomial ànot the difference of two squares…
The GCF is 3y²
3y²(9 – 16y²)
WOW—Now we have the DIFFERENCE of TWO SQUARES
3y²(3 + 4y)(3 - 4y)

Called FACTORING COMPLETELY because you cannot factor any further
Always check your factoring by distributing or FOILing back!BTW…  There is NO SUCH THING as the Sum of Two Squares!

a² + b² CANNOT be factored!!

but… -b² + a² is really a² –b² because it is just switched ( commutative)

Algebra Honors

Special Products 8-4

Learning to recognize some special products- it makes multiplying easier

Remember: you can FOIL these just like the other products until you remember these special patterns  but when we get to factoring  it really helps you to know these patterns by heart!!

When you do, you actually don’t need to show any work because you do it in your head!  That should make a lot of you very happy !!

Difference of Two Squares

You will notice that the two factors are Identical except that they have different signs
(x + 6)(x – 6) = x² – 6x + 6x – 36 = x²  – 36  The middle terms drop out.

This will happen EVERY TIME
The middle terms are additive inverses so they become ZERO
You are left with a difference (subtraction) of two terms that are squared.

Squaring a Binomial

When you multiply one binomial by itself (squaring it) you end up with:
First term squared + twice the product of both terms + last term squared

(x + 6)² = ( x + 6)(x + 6) = x²+ 2(6x) + 36=  x²+ 12x + 36

If you had foiled it
x²+ 6x + 6x + 36 à can you see that the 2 middle terms are just doubling up?

Another Example
(x - 6)² = ( x - 6)(x - 6) = x² -2(6x) + 36=  x² - 12x + 36

PLEASE NOTE:  Noticing these special products helps you do these multiplication FASTER
If you ever forget them, just FOIL ( or box)
You will need to start recognizing them for factoring!!

Algebra Honors

CHAPTER 8-3: MULTIPLYING POLYNOMIALS

MULTIPLICATION OF A BINOMIAL BY A TRINOMIAL

OR TWO TRINOMIALS

When you multiply a trinomial by a binomial or two trinomials,

 it gets really tricky!

2 ways:
1) box method
2) column method
(double or triple distributive with columns to combine like terms)
I’LL SHOW YOU THESE METHODS IN CLASS

If you have a trinomial times a binomial, it's easier to use the Commutative Property
and make it a binomial times a trinomial:
(x² + x - 1) (x - 1)
switch it to
(x - 1) (x² + x - 1)

Algebra Honors

Adding and Subtracting Polynomials 8-1

VOCABULARY

Monomial: a constant, a variable or product of variables and WHOLE number exponents – one term

Binomial: The SUM of TWO monomials

Trinomial: The SUM of THREE monomials

Polynomial: a monomial or the sum of monomials

Degree of monomial: The sum of the exponents of all the variables in the term

Degree of polynomial: The highest degree of any term in a polynomial

Standard form  (Descending form): Place variables in alphabetical order with the highest power  first

Leading coefficient: The coefficient of the term that has the highest degree.

Degree	Name
0	constant
1	linear
2	quadratic
3	cubic
4	quartic
5	quintic
6	6th degree

ADDING AND SUBTRACTING POLYNOMIALS
This is nothing more than combining LIKE TERMS so this is DÉJÀ VU!

LIKE TERMS = same variable AND same power

You can do this using 3 different strategies:
1. Simply do it in your head, but keep track by crossing out the terms as you use them.
2. Rewrite, putting the like terms together (commutative and associative property)
3. Rewrite in COLUMN form, putting like terms on top of each other like you do when adding a column of numbers.

EXAMPLE OF COLUMN FORM:
[5x4 - 3x2 - (-4x) + 3] + [-10x4 + 3x3- 3x2 - x + 3]

Rewrite in column form, lining up like terms:
LEAVE A SPACE IF ONE OF THE POWERS ARE MISSING AND MAKE SURE BOTH OF THEM ARE IN DESCENDING ORDER!

                                                                                                                                          

                                5x4           - 3x2 - (-4x)  + 3                                                                        

                         + -10x4  + 3x3 - 3x2 -     x + 3                                                                                 

                           -----------------------------------                               

                              -5x4  + 3x3 - 6x2  + 3 x  + 6

SUBTRACTING POLYNOMIALS
You can use the ADDITIVE INVERSE PROPERTY (our BFF) with polynomials!
Subtracting is simply adding the opposite so.............
DISTRIBUTE THE NEGATIVE SIGN TO EACH TERM!!
(Change all the signs of the second polynomial!)
After you change all the signs,

use one of your ADDING POLYNOMIAL strategies!

EXAMPLE OF COLUMN FORM:
[5x4 - 3x2 - (-4x) + 3] - [-10x4 + 3x3- 3x2 - x + 3]
Rewrite in column form, lining up like terms:

                                                                                                                          

      5x4          - 3x2 - (-4x) + 3

- ( -10x4 + 3x3 - 3x2 -      x + 3)

        -----------------------------------
                                                                                                 

DISTRIBUTE THE NEGATIVE, THEN ADD:
                                                                                                                         

     5x4          - 3x2 -  (-4x) + 3   

+10x4 - 3x3 + 3x2 +      x - 3

                                ----------------------------------- 15x4 - 3x3    + 5 x

Math 6A ( Period 1 & 2)

Comparing & Graphing Ratios 5-4

Working with your partners, you complete the following ratio table for each frosting mixture. In order to complete these accurately,  you had to listen to what the students’ frosting recipe called for as well as what my teacher recipe called for.

After you have completed the Drops of Blue column, label the third column that will show the total number of drops for each relationship.  Then complete that column as well.

Whose frosting is bluer? Whose frosting is redder? Justify your answers.

Students’	Frosting			Teacher’s	Frosting
Drops of Red	Drops of Blue			Drops of Red	Drops of Blue
1				3
2				6
3				9
4				12
5				15

We then to use the value from the ratio table for your frosting to create a graph in the coordinate plane. Together, as a whole class we used the values in the table to plot the points. Then connected the points and each of you discussed the graph with your partners. What do you notice?

We continued and working with your partners each of  you  completed the following table and the graph

Teacher’s	Frosting
Drops of Red	Drops of Blue
3
6
9
12
15

You explained the relationship between the entries in the ratio tables and the points on the graph. How is this graph similar to the first graph?  How is it different?

How can you use the graphs to determine whose frosting has more red or blue in it? You and your partners discussed this question.