Algebra (Period 1)

Addition of Polynomials 5-7
This is nothing more than combining LIKE TERMS
LIKE TERMS = same variable AND same power

You can either 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:


Subtraction of Polynomials 5-8
You can use the ADDITIVE INVERSE PROPERTY 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!
(see the 3 strategies listed above under Chapter 5-7)

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)
-----------------------------------

For the sake of showing you here, I have added ZERO Terms to line up columns
+ 5x⁴ + 0x³ - 3x² -(-4x) + 3
-(-10x⁴ +3x³- 3x² - x + 3)
-----------------------------------

DISTRIBUTE THE NEGATIVE, THEN ADD:
5x⁴ + 0x³ - 3x² - (-4x) + 3
+10x⁴ -3x³ +3x² + x - 3
-----------------------------------
15x⁴ - 3x³ + 5 x

Pre Algebra (Period 2 & 4)

Solving 1 Step Inequalities with Addition & Subtraction 2-9


These are solved just like equations except you have a greater than, less than, greater than or equal, less than or equal sign instead of the equal sign.


Your goal is still the same:

ISOLATE THE VARIABLE

You sitll balance with the INVERSE OPERATION

Your justifications are still almost the same:
IDENTITY AND PROPERTIES OF INEQUALITY



Solving 1 Step Inequalities with Multiplication & Division 2-10

These are solved just like equations except you have a greater than, less than, greater than or equal, less than or equal sign instead of the equal sign.


Your goal is still the same: ISOLATE THE VARIABLE


You sitll balance with the INVERSE OPERATION


Your justifications are still almost the same:
IDENTITY AND PROPERTIES OF INEQUALITY



THERE IS ONE MAJOR EXCEPTION!!!!

When you multiply or divide by a NEGATIVE, 
the symbol CHANGES DIRECTION!


-3y > 9

You need to divide both sides by NEGATIVE 3 so the symbol will switch from > to < in the solution
 y < -3 is the answer

 If you want to understand why:
 3 < 10 correct? 
Now multiply both sides by -1 (mult prop of equality)
 You get -3 < -10, but THAT'S NOT TRUE!!! 
You have to SWITCH THE SYMBOL to make the answer true: -3 > -10

Algebra (Period 1)

Polynomials 5-5

Polynomials = SUM of monomials

Monomials must have variables with whole number powers. Review section 5-3 for detail on monomials!!
no variables in the denominator, no roots of numbers!!
so 1/x is not a monomial
neither is x ^1/2

constants have whole number power of zero..
7 is really 7x⁰

1 term = monomial
2 terms = binomial
3 terms = trinomial

TERMS are separated by addition
( if see subtraction-- THINK: add the opposite!!)

Coefficient - number attached to the variable ( it can be a fraction)
3x² - 10x
the coefficients are 3 and -10. Make sure to attach the negative sign to the coefficient -- and ADD the OPPOSITE

y/6 is really (1/6)y so the coefficient is 1/6
if you have -x/3 that is really (-1/3)x so the coefficient is -1/3


Constant = the number that is not attached to ANY variable

Some TERMS YOU NEED TO KNOW

Degree of a term = SUM of the exponents of all its variables
-6x⁴ : the degree is 4
8x² : the degree is 2
-2x : the degree is 1
9 : the degree is 0 ( think 9 is really 9x⁰

Degree of a polynomial - HIGHEST degree of any of its terms
so
-6x⁴ + 8x² + -2x + 9
The degree of the polynomial is : 4

Leading term = term with the HIGHEST degree
Leading coefficient- the coefficient of the leading term




More on Polynomials 5-6

Descending order- write the variables with the highest power first ( This is the way it is usually written)

Ascending order- write the variables with the lowest powerr first ( actually NEVER used in practice)

Evaluating a polynomial- this is what we have been doing all year... plug it in, plug it in!!
Remember to ALWAYS put the number you substitute in parentheses!!

2x²y + 5xy - 4, where x = -4 and y = 5
Substitute carefully:

2(-4)²(5) + 5(-4)(5) - 4
= 2(16)(5) +(-20)(5) - 4
= 160 +(-100) -4
= 160 -104
= 56

Pre Algebra (Period 2 & 4)

Inequalities & Their Graphs 2-8

What most real life situations call for

How often to you think...I need at least $20 (not exactly $20)
....
I want at most 20 minutes of homework (not exactly 20 minutes)?


Inequalities have many answers (most of the time an infinite number!)

Examples: n > 3 means that every real number greater than 3 is a solution! (but NOT 3)

n ≥ 3 means still means that every real number greater than 3 is a solution, 
but now 3 is also a solution


n < 3 means that every real number less than 3 is a solution! (but NOT 3)
 n ≤ 3 means still means that every real number less than 3 is a solution,   but now 3 is also a solution

 GRAPHING INEQUALITIES:
 First, graphing an equation's solution is easy
 1) Say you found out that y = 5, you would just put a dot on 5 on the number line
 2) But now you have the y ≥ 5
 You still put the dot but now also darken in an arrow going to the right showing all those numbers are also solutions
 I say: the equal sign part of it is a crayon...Pick up the crayon and color in the circle
 3) Finally, you find in another example that y > 5

You still have the arrow pointing right, but now you OPEN THE DOT on the 5 to show that 5 IS NOT A SOLUTION!

I say: there isn't a crayon so you can't color in the circle.
Therefore it stays open!
 

TRANSLATING WORDS:

Some key words to know:


AT LEAST means greater than or equal that is...
≥

AT MOST means less than or equal. THat is ≤


I need at least $20 to go to the mall means I must have $20, but I'd like to have even more!


I want at most 15 minutes of homework means that I can have 15 minutes, but I'm hoping for even less!