1. Polynomials

2. 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

3. 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.

4. 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

5. 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

6. 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

7. POLYNOMIAL
Monomial
Binomial
Trinomial
2x + 9 x 3
10x2 + 2x + 9
2(x + 4)
3x + 4
6x - 8
-9x
3x2 + 3xy + 9x 10 2x
x2 + 3xy + 9xyz

11. Algebra Tiles & Area x
x2 - tile x 1
x-tile x
1-tile 1
1-tile 1

12. Draw algebra tiles to represent the polynomial 3x2 – 2x + 5
Recall This Algebraic Expression has 3 Terms:
3x is the coefficient, x2 is the variable
–2x is the coefficient, x is the variable
is the constant term

13. 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

14. Length = 5 This rectangle has the following properties: Width = x
We can combine algebra tiles to form a rectangle. We can then write the area and the perimeter of the rectangle as a polynomial.
This rectangle has the following properties:
Length = 5
Width = x
Perimeter is x x + 5 = 2x + 10
Area = LW = 5 x x = 5x x 5

15. Determine the Area & Perimeter of the following Rectanglesx x x x
This rectangle has the following properties:
Length = 3x
Width = x
Perimeter = x + 3x + x + 3x = 8x
Area = (3x) * (x) = 3x2

16. 1.) 3.) 2.) Length ________ Width ________
Area ______________________
Perimeter __________________
Width ________
Length ________
3.)
1.)
2.)

17. 2.) Length x Width 2 Perimeter x + x + 2 + 2 = 2x + 4
Area (4) * (x) = 4x
Perimeter x x + 4 = 2x + 8
Width x
Length 4 3)
Area (2x) * (2x) = 4x2
Perimeter 2x + 2x + 2x +2x = 8x
Width 2x
Length 2x
Area (2) x (x) = 2x
Perimeter x + x = 2x + 4
Width 2
Length x 1)
2.)

18. 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:

19. Multiplying Monomials
(3x2)(5x3) = (3 * x * x) (5 * x * x * x)
= (3) (5) (x*x*x*x*x)
= 15x5

20. With Algetiles
x * x = x2
(2)(5x) = 10x

21. So 3 x 2 x 2 are prime factors of 12
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
FACTOR
TREES 4 3 2 6 2 3
So 3 x 2 x 2 are prime factors of 12

22. Practice Exercises:
Express each number as a product of its prime factors: 30 36 25 42 75
100
121
150

23. Practice Solutions: 2 x 3 x 5 2 x 2 x 3 x 3 5 x 5 2 x 3 x 7 3 x 5 x 5

24. We can factor in algebra too
3x2 = 3 * x * x
5x = 5 * x
2x4 = 2 * x * x * x * x
2x2y2 = 2 * x * x * y * y
Let’s Try:
a)4x3 b) –x c)2x6
d) 9x2y e) -6a2b2

25. 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)

26. 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.

27. List the common prime factors in each list: 2, 3.
Finding the GCF
To find the GCF of 126 and 60.
= 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.

28. List the common prime factors in each list:
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

29. Finding the GCF Try These Together 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.

30. Finding the GCF What is the GCF of 14 and 20? Factors of 14 = 2, 7
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