Day 3: Daily Warm-up.
Simplify each expression (combine like terms)
1.
4
2
4
2
2.
 9x  4x 1 3x  2x  3
5x 13x  x x  x  3x
3
2
3
2
Find the product and combine like terms.
3.
x  2
4.
x  3x  3
5.
5x  32x  1
2
A.3a Polynomials and Factoring
Special Products and Factoring
A.3 Objectives
A. Polynomials and Factoring
1. Understand the vocabulary of polynomials
2. Add and subtract polynomials
3. Write polynomials in standard form.
4. Multiply polynomials
5. Special Products
6. GCF
7. Factor Polynomials with 2 or 3 terms.
8. Factor Polynomials with more than 3 terms.
9. Find the GCF of any polynomial expression
1. Vocabulary of Polynomials
Monomial:
the product of a constant and a variable raised to a
nonnegative integer power.
ax
coefficient of the monomial
(constant value)
k
variable
degree of the
monomial (power
of the variable)
Polynomial: sum or difference of monomials

3x 2  4 y  2x  6  5x 2  3y 2  3x 1
1. Vocabulary of Polynomials
Special Polynomial Names
if # terms is:
Name:
2 terms
Binomial
3 terms
Trinomial
Degree of Polynomial
if Highest degree term is:
degree 1
degree 2
degree 3
Name:
Linear
Quadratic
Cubic
Like terms : contain same power of the variable.
2. Add/Subtract Polynomials
Add the coefficients of
like terms(same power of the variable)
Examples

 

1)  9 x  4 x  1  3x  2 x  3
2)
5x
4
3
2
 
4
2
 13x  x  x  x  3x
2
3
2

3. Standard Form and Degree
Definition
Standard Form:
polynomial written in descending order
Example
Write the standard form for:
Definition
3x  x  4 x  7
4
2
The Degree of a polynomial:
is the degree of the greatest degree term.

Example
What is the degree of this polynomial?
x  4
2
4. Multiplying Polynomials
1. Double Distribution
•Using the distributive property, we can multiply any
number of terms
3x  2x
2

 4x 1
2. Foil Method
•Special case: only works with two binomials
x  3x  2 
4. Multiplying Polynomials
Examples for YOU to try….
Find the product and write in STANDARD form


1) x  1 x  x  1
2
2) (3x  1)  y (3x  1)  y 
3) 2 x  1x  21  x
5. Special Products
Study Tip
These are products that occur often. You should know these!
Difference of Two Squares.
 A  B A  B
 A B
2
Perfect Square Binomial.
A  B
 A  2 AB  B
2
A  B
 A  2 AB  B
2
2
2
2
2
2
6. GCF (Greatest Common Factor)
Example. Factor out the GCF
4 x  2 x  16 x
3
2
GCF of coefficients:
Largest number that divides into all coefficients.
GCF of variable expressions:
Find smallest exponent
Definition of Prime:
If a polynomial does not factor into 2 or more polynomials, it is Prime.
Day 4: Daily Warm-up.
x  1  x  1
2
1. Simplify completely.
2. State the domain of
2
1
2x  4
 1
3. Simplify (no negative exponents)
3

xy z

2
6x y z
4. Factor completely.
3x  12 x  36 x
3
2

2 3 2

2
7. Factoring Polynomials
Definition: Factoring is writing a polynomial as a
product of polynomials of lower degree
General Steps for Factoring:
1. Factor out GCFs
2. Is it a binomial (with no middle term) ?
a) Is it a Difference of squares or sum/difference of cubes
A2  B 2 or A3  B 3 or A3  B3
3. Is it a trinomial (only 3 terms)
• apply factorization algorithm
4. If more than 3 terms
• grouping
5. More Special Products
Difference of Two Cubes.
 A  B A
2

 AB  B 
2
Sum of Two Cubes.
 A  BA
2

 AB  B 
2
A B
3
3
A B
3
3
A. Factoring Binomials
Study Tip
Factor each. If it does not factor, state that it is PRIME.
1. x 2  25
2.
4 x  25
3.
x 125 
4
3
4.
8x  1
5.
x 1
3
2
2
B. Factoring Trinomials: ax
Simple case, when a  1
Example 0.
 bx  c
x  4 x  12
2
1. Factor out GCF, if there is one.
c
2. What multiplies to c and adds
up to b?
3. Rewrite as:
(x
b
)( x
)
C. Factoring Algorithm for case when a  1
ax  bx  c
2
Example 1.
10 x  x  3
2
1. Factor out GCF.
2. Multiply a and c
ac  10(3)  30 b  1
3. Find the factors of ac
that add to b.
4. Rewrite the middle term
bx as sum of the 2 factors.
5. Grouping.
-Double bubble (check signs)
-Factor out GCF in each group, if no GCF, write 1.
Factoring Algorithm
ax  bx  c
2
Example 2.
6 x  19 x  7
2
1. Factor out GCF.
2. Multiply a and c
3. Find the factors of ac
that add to b.
ac 
4. Rewrite the middle term
bx as sum of the 2 factors.
5. Grouping.
-Double bubble (check signs)
-Factor out GCF in each group
b
Factoring Algorithm for simple case,
a 1
x  bx  c
2
Example 3.
x  4 x  12
2
1. Factor out GCF.
c
2. What multiplies to c and adds
up to b?
3. We can skip rewrite of the
middle term. (Why?)
(x
b
)( x
Factor this polynomial using the “double bubble” algorithm.
Do you get the same result?
)
Practice Time!
completely factor the polynomial
1.
x x
2.
8 x 2  32 x  30
3.
x  2x 1
4.
x5  x 2
7
6
5
3
7. Polynomial with 4 terms
Use Grouping (double bubble)
Ex. 3x 3  2x 2 12x  8
Factoring Practice
completely factor the polynomial
1.
4 x  20x  9x  45
2.
20x 6 12x 5  35x 3  21x 2
3
2
8. Finding the GCF of an expression
1. 6 x(2  x)  9 x 2  x 
4
2.
2
3
3x  4  2 x  33x  4
2
5. More Special Products
You may wish to memorize these, but could also derive them.
Cubes of Binomials, or Perfect Cubes.
A  B
3
A  B
3


A  3 A B  3 AB  B
3
2
2
3
A  3 A B  3 AB  B
3
2
2
3