Unit 9: Polynomials and Factoring

(Also known as a term)
An expression that is one of the following
a number
a variable
the product of numbers and/or variables
Ex: -abc
The addition and subtraction of monomials
Example: y^2 + 7y - 9
2 term polynomial
Example: 5x + 7
3 term polynomial
Example: y^2 + 7y - 9
Degree of a monomial
The sum of the exponents of variables
5x degree: 1
6x^3y^2 degree: 5
Degree of a polynomial
The largest exponent
3x^4 + 5x^2 - 7x + 1
Degree = 4
Descending Order
Variables should be written in alphabetical order
When there is more than one term with the sane term with the same variable, the term with the largest exponent will appear first
Variable must be in alphabetical order and within the same variable, must be written in descending order

Example: 14c^3 - c^2 + 2c +16
Adding Polynomials
CLT = Combining Like Terms
1) Add the coefficients
2) Keep the variable and their exponents exactly the same
(17x^3 - 4x^2 - 3x) + (2x + 5x^2 + 4x^3) = 21x^3 + x^2 + x
Subtracting Polynomials
1) Add the opposite of all terms in the parentheses that appear after the subtraction sign. Note: this is the same thing as distributing the negative to all of the terms that appear after the subtraction sign

2) Follow the adding polynomials rules

(9y^2 - 3y + 4) - (5y + 6) = 9y^2 -8y - 2
Multiplying Polynomials
There are two main techniques:
1) FOIL 2) BOX (chart)
a shortcut for the distributive property and can be used only when you multiply two binomials
F multiply the first terms of each binomial
O outer
I Inner
L last
Ex. (y + 3) (y + 7)
y^2 + 7y + 3y + 21
y^2 + 10y + 21
BOX Method
The box method works for all types of polynomial multiplication

Steps for the chart (box) method:
1) What are the dimensions of the chart
2) Draw and label the chart with each term
3) Multiply the terms and fill in the products in the individual boxes
4) Add the boxes of like terms (CLT) to determine the final answer
5) Write answers in descending order
Dividing Polynomials by Monomials
1) divide each term in the numerator (dividend) by the monomial denominator(divisor)
Note: Assume no denominator/divisor = 0

2) simplify completely

Ex: 12x-9y-3/3

Follow order of operations (divide first then + or -)
When subtracting, subtract the entire answer of the second division problem

Ex: 3x-6/3=4-8x/2
Degrees of Monomials and Polynomials
Use Zoom 6
The degree of a monomial is the sum of the exponents of all its variable
The degree of a polynomial is the largest degree of any of its terms after it has been simplified
Ex: 4x^2-11x-5
Greatest Common Factor: Gotta Check First
Prime number: A # with only 2 factors, 1 and itself.
Composite: A # with 3 or more factors
Note: 1 is NOT prime and 1 is NOT composite because 1 has only 1 factor
Greatest Common Factor
The biggest number that is a factor of the 2 or more terms. It is found by finding the product of the common prime numbers

prime: 2, 3, 5, 7, 9, 11, 13, 17, 19
What is factoring?
Factoring is the process of converting an algebraic expression to a multiplication problem (or a product of factors)

Polynomial ---- Multiplication Problem

It's the reverse of multiplication!

We start with a polynomial and then through the process of factoring, we end up with the product of two (or more) factors.

Factoring is when we break down the polynomial by stripping away common factors....or factors that can divide evenly into two or more values. We can factor numbers and variables
Factoring Trinomials using the X-BOX Method!
ax^2 + or - bx + or - c

Check for the GCF first

Follow steps on sheet
Factoring Trinomials using the Grouping Method
1. Multiply the numbers found in the quadratic term and the constant term. (a x c)
2. Using the circle, find the Magic Pair
3. Re-write the "middle man" as the sum/diff of the magic pair (how FOIL would be written out(
4. Group the first two terms and the last two terms
5. Factor out the GCF in each pair/set of terms
6. Factor out the common factor/parenthesis and put the left-overs in another set of parenthesis
7. Check by FOIL
Difference of Squares
Check to see if the binomial is a Difference of Squares
a) Are BOTH terms perfect squares?
b) Is there a subtraction sign between the two terms

If YES to both a and b, than determine the answer
1) Parenthesis-Write 2 sets ( ) ( )
2) Signs-one ( ) will be +,one ( ) will be -
3) 1st term-Take square root of 1st term in the problem
4) 2nd Term-Take square root of 2nd term in the problem
5) Check by FOIL. Multiply your answer. The result should be the original problem.
Dividing Polynomials by Binomials
Factoring Method
1) Factor the polynomial(s)
2) Easier to write problem in fractional format
3) Wipe out/cancel common factor groups