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Polynomials
In This Unit…
 Review simplifying polynomials, distributive property &
exponents
 Classifying Polynomials
 Area & Perimeter with Algetiles
 Factoring in Algebra
 Multiplying & Dividing Monomials
 Multiplying Polynomials & Monomials
 Factoring Polynomials
 Dividing Polynomials
 Multiplying Two Binomials
 Factoring Trinomials
 A monomial is a number, a variable, or a product of
numbers and variables.
 A polynomial is a monomial or a sum of monomials. The
 exponents of the variables of a polynomial must be
positive.
 A binomial is the sum of two monomials, and a trinomial
is the sum of three monomials.
 The degree of a monomial is the sum of the exponents of
its variables.
 To find the degree of a polynomial, you must find the
degree of each term. The greatest degree of any term is
the degree of the polynomial. The terms of a polynomial
are usually arranged so that the powers of one variable
are in ascending or descending order.
Classifying Polynomials
A monomial is an expression with a single
term. It is a real number, a variable, or the
product of real numbers and variables.
Example: 4, 3x2, and 15xy3 are all monomials
Classifying Polynomials
A binomial is an expression with two terms. It
is a real number, a variable, or the product of
real numbers and variables.
Example: 3x + 9
Classifying Polynomials
A trinomial is an expression with three terms.
It is a real number, a variable, or the product
of real numbers and variables.
Example: x2 + 3x + 9
Now you try to Classify Each
POLYNOMIAL
Monomial
x
2
x
10x + 2x + 9
2(x + 4)
x
3x + 4
x
6x - 8
x
-9x
x
2
x
3x + 3xy + 9x
2
10
x
2x
x
x + 3xy + 9xyz
Trinomial
x
2x + 9
3
Binomial
x
Algebra Tiles & Area
x
x
x2 -
tile
1
x-tile
x
1
1
1-tile
1-tile
Draw algebra tiles to represent the polynomial
3x2 – 2x + 5
Recall This Algebraic Expression has 3 Terms:
3x2
3 is the coefficient, x2 is the variable
–2x
-2 is the coefficient, x is the variable
5
5 is the constant term
What is the area of a rectangle?
AREA = Length x Width
How do you find the perimeter of a rectangle?
ADD up all of the sides
We can combine algebra tiles to form a rectangle.
x
We can then write the area and the perimeter of the rectangle as a
polynomial.
5
This rectangle has the following
properties:
Length = 5
Width = x
Perimeter is x + 5 + x + 5 = 2x + 10
Area = LW = 5 x x = 5x
Determine the Area & Perimeter
of the following
Rectangles
x
x
x
x
This rectangle has the following properties:
Length = 3x
Width = x
Perimeter = x + 3x + x + 3x = 8x
Area = (3x) * (x) = 3x2
Length ________
1.)
Width ________
Perimeter __________________
Area ______________________
2.)
Length ________
Width ________
Perimeter __________________
Area ______________________
3.)
Length ________
Width ________
Perimeter __________________
Area ______________________
Length x
1)
Width 2
Perimeter
(2) x (x) = 2x
Area
2.)
3)
x + x + 2 + 2 = 2x + 4
Length
2x
Width
2x
Perimeter
2x + 2x + 2x +2x = 8x
Area
(2x)
Length
4
Width
x
* (2x) = 4x2
Perimeter x + 4 + x + 4 =
Area
(4) * (x) = 4x
2x + 8
Multiplying Monomials
RECALL :
 Multiplying Powers: When multiplying powers
with the same base we add the exponents
Example: x2 * x2 = x4
 Dividing Powers: When dividing powers with
the same base we subtract the exponents
Example: x3 ÷ x1 = x2
 Power of a Power:
Example:
Multiplying Monomials
(3x2)(5x3)
= (3 * x * x) (5 * x * x * x)
= (3) (5) (x*x*x*x*x)
= 15x5
With Algetiles
x * x = x2
(2)(5x) = 10x
Prime Factor Review
A prime factor is a whole number with
exactly TWO factors, itself and 1
A composite number has more than two
factors 12
12
3
3
FACTOR
TREES
4
2
2
6
2
2
2
So 3 x 2 x 2 are prime factors of 12
3
Practice Exercises:
Express each number as a product of its prime
factors:
a)
b)
c)
d)
e)
f)
g)
h)
30
36
25
42
75
100
121
150
Practice Solutions:
a)
b)
c)
d)
e)
f)
g)
h)
2x3x5
2x2x3x3
5x5
2x3x7
3x5x5
2x2x5x5
11 x 11
2x3x5x5
We can factor in algebra too
1) 3x2 = 3 * x * x
2) 5x = 5 * x
3) 2x4 = 2 * x * x * x * x
4) 2x2y2 = 2 * x * x * y * y
Let’s Try:
a)4x3
d) 9x2y
b) –x2
e) -6a2b2
c)2x6
We can factor in algebra too
a)4x3 = 4 * (x * x * x )
b) –x2 = (-1) * (x * x)
c)2x6 = (2) * (x * x * x * x * x * x)
d) 9x2y = (9 * 2) * (x) * (y)
e) -6a2b2 = (-6) * (a * a) * (b * b)
Greatest Common Factor
The greatest of the factors of two or more
numbers is called the greatest common
factor (GCF).
Two numbers whose GCF is 1 are
relatively prime.
Finding the GCF
To find the GCF of 126 and 60.
126 = 2 x 3 x 3 x 7
60 = 2 x 2 x 3 x 5
List the common prime factors in each list:
2, 3.
The GCF of 126 and 60 is 2 x 3 or 6.
Finding the GCF
Find the GCF of 140y2 and 84y3
140y2 = 2 * 2 * 5 * 7 * y * y
84y3 = 2 * 2 * 3 * 7 * y * y * y
List the common prime factors in each list:
2, 2, 7, y, y
The GCF is 2 * 2 * 7 * y * y = 28y2
Finding the GCF
Try These Together
1.What is the GCF of 14 and 20?
2. What is the GCF of 21x4 and 9x3?
HINT: Find the prime factorization of the numbers and then find the product of
their common factors.
Finding the GCF
1.What is the GCF of 14 and 20?
Factors of 14 = 2, 7
Factors of 20 = 2, 4, 5, 10
Therefore the GCF is 2
2. What is the GCF of 21x4 and 9x3?
Factors of 21x4 = 3 * 7 * x * x * x * x
Factors of 9x3 = 3 * 3 * x * x * x
Therefore the GCF is 3* x * x * x = 3x3