Algebra Period 3 (Review)

REVIEW:WEEK BEFORE WINTER BREAK
Sections 5-5 to 5-8 Polynomials

Polynomial = sum of monomials

Monomials must have variables with WHOLE NUMBER powers
(constants have whole number powers because you can say it has a variable to the zero)
(no variables in the denominator and no roots of numbers)
1 Term = monomial
2 terms = binomial
3 terms = trinomial

Terms are separated by addition
(and subtraction...although we never subtract...we add the opposite.)

Coefficient = Number attached to variable (can be a fraction!)
The sign of the coefficient should be looked at AFTER you change any subtractions to addition
For example: 3x² - 10x
The coefficients are 3 and -10
Also, if you have y/6, you really have 1y/6 (ID prop of multiplication)
so the coefficient of y/6 is 1/6 and can be written as (1/6) y
if you have -y/6, you really have -1y/6 (ID prop of multiplication)
so the coefficient of -y/6 is -1/6 and can be written as (-1/6) y

Constant = the number that is not attached to any variable

Degree of a term = sum of the exponents of all its VARIABLES
Degree of a polynomial = HIGHEST degree of any of its terms
Leading term = term with the HIGHEST degree
Leading coefficient = the coefficient of the LEADING TERM

Sections 5-6
DESCENDING ORDER - Write the variables with the highest power first
ASCENDING ORDER - Write the variable with the lowest power first
(this order is actually never used in practice!)
EVALUATING A POLYNOMIAL - WE'VE BEEN DOING THIS ALL YEAR! Plug it in, plug it in, plug it in! Then use Aunt Sally!
Remember to ALWAYS put the number you substitute in parentheses!!!


Sections 5-7 ADDING POLYNOMIALS
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:

Section 5-8: SUBTRACTING POLYNOMIALS
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