March 18, 2014

2. The sum of 2 binomials is 5x2 - 6x. If one of the binomials is 3x2 - 2x, what is the other binomial?

1. 5(4x - 4) - 3 = 372

3. Solve in 1 Step: What is the sale price of a $72 pair of shoes discounted 20%?

Warm-up:(5)

4. A 20% profit was made on an item selling for $60. What was the cost of the item?

Warm-up:(5)

5. Bruce likes to amuse his brother by shining a flashlight on his hand and making a shadow on the wall. How far is it from the flashlight to the wall?

Class Notes Section of Notebook, please

Dividing Polynomials(2)

x4 - 81 ÷ x - 3The pattern is: divide, multiply, subtract, repeat.

2x3 – 13x2 + 26x – 24 (x – 4)

Factoring Polynomials

The point of factoring is to simplify

Factors: Quantities that are multiplied together to form a product.

3

Factors Product

An algebra example:

(x + 2)(x + 3) =

Factors Product

x2 + 5x + 6

There are several methods that can be used when factoring polynomials. The method used depends on the type of polynomial that you are factoring.

We will spend the next few weeks learning to factor by:

1. Greatest Common Factor 2. Grouping 3. Difference of Squares

4. Sum or Difference of Cubes 5. Trinomials 6. Special Cases


** Remember that the method of factoring depends on the type of polynomial being factored. Throughout this process, pay attention not only on how to factor, but the type of polynomial being factored. As we progress, you will have to correctly match the factoring method with the polynomial.

Grouping Example: 3ax + 6ay + 4x + 8y; No obvious factor

3a(x + 2y) + 4(x + 2y); The factored form is: (x + 2y) (3a+ 4)

Difference of two Squares; Example: 9x2 - 25y4

The fact that there's no middle term tells us the signs of the binomial factors must be:

The factored form is: (3x +5y2 )( 3x- 5y2)

( ) ( )

( + )( - );

Trinomials with leading coefficient of 1: x2 + bx + c

Example: x2 – 5x + 6


We are looking for two numbers whose sum is -5 and whose product is 6. You can make a table of factors for 6, and see which, if any, numbers fit. Factors of: 6

1,6

2,3Let's try one more:

36

1,36

2,18

3,12

4,9

6,6

Whole numbers that are multiplied together to find a product are called factors of that product. A number is divisible by its factors.

Greatest Common Factor

2•2 •3 =12

We will begin with Factoring the Greatest Common Factor

Pros: -- simple to understand

Cons: -- most polynomials cannot be factored this way

Our goal, whether factoring numbers or polynomials, is to take out the GCF thus simplifying the as much as possible. To make sure we have the GCF, prime factorization is utilized.

A prime number has exactly two factors, itself and 1. The number 1 is not prime because it only has one factor.

Remember!

Factors; GCF

Example 1: Writing Prime Factorizations

Write the prime factorization of 98.

Method 1 Factor tree Method 2 Ladder diagram

Choose any two factors of 98 to begin. Keep finding factors until each branch ends in a prime factor.

Choose a prime factor of 98 to begin. Keep dividing by prime factors until the quotient is 1.

98 = 2 7 7

98

49

7

1

2

7

7

98 = 2 7 7

The prime factorization of 98 is 2 7 7 or 2 72

98

2 49

7 7

Factors; GCF

Write the prime factorization of each number.

a. 40

40

2 20

2 10

2 5

333

11

b. 33

40 = 23 5 33 = 3 11

The prime factorization of 40 is 2 2 2 5 or 23

5.The prime factorization of 33 is

3 11.

Factors; GCF

Factors that are shared by two or more whole numbers are called common factors. The greatest of these common factors is called the greatest common factor, or GCF. Find the GCF of 12 and 32.

Factors of 12: 1, 2, 3, 4, 6, 12

Factors of 32: 1, 2, 4, 8, 16, 32

Common factors: 1, 2, 4

The greatest of the common factors is 4.

Factors; GCF

Find the GCF of each pair of numbers.

100 and 60

factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100

factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

The GCF of 100 and 60 is 20.

List all the factors.

Circle the GCF.

Method 1 List the factors.

Factors; GCF


26 and 52

26 = 2 13

52 = 2 2 13 Align the common factors.

2 13 = 26


Method 2 Prime factorization.

Factors; GCF


15 and 25

25 = 1 5 5


Align the common factors.1 5 = 5

15 = 1 3 5

Method 2 Prime factorization.

Factors; GCF

You can also find the GCF of monomials that include variables.

To find the GCF of monomials:1.) Write the prime factorization of each coefficient 2.) Write all powers of variables as products. 3.) Find the product of the common factors.

Factors; GCF

Example 3A: Finding the GCF of Monomials

Find the GCF of each pair of monomials.

15x3 and 9x2

15x3 = 3 5x x x

9x2 = 3 3•x x

3 x x = 3x2

Write the prime factorization of each coefficient and write powers as products.

Align the common factors.

The GCF of 3x3 and 6x2 is 3x2.

Factors; GCF

Example 3B: Finding the GCF of Monomials


8x2 and 7y3

8x2 = 2 2 2 x x

7y3 = 7 y y y



There are no common factors other than 1.

The GCF 8x2 and 7y is 1.

Factors; GCF

If two terms contain the same variable raised to different powers, the GCF will contain that variable raised to the lower power.

Helpful Hint

Factors; GCF

Example 3a


18g2 and 27g3

18g2 = 2 3 3 g g

27g3 = 3 3 3 g g g

3 3 g g

The GCF of 18g2 and 27g3 is 9g2.


Find the product of the common factors.

Factors; GCF

Example 3b


16a6 and 9b

9b = 3 3 b

16a6 = 2 2 2 2 a a a a a a


There are no common factors other than 1.

The GCF of 16a6 and 7b is 1.

Factors; GCF

Example 3c


8x and 7v2

8x = 2 2 2 x

7v2 = 7 v v



There are no common factors other than 1. The GCF of 8x and 7v2 is 1.

Factors; GCF

Application of the GCF:

A cafeteria has 18 chocolate-milk cartons and 24 regular-milk cartons. The cook wants to arrange the cartons with the same number of cartons in each row. Chocolate and regular milk will not be in the same row. How many rows will there be if the cook puts the greatest possible number of cartons in each row?

The 18 chocolate and 24 regular milk cartons must be divided into groups of equal size. The number of cartons in each row must be a common factor of 18 and 24.

Factors; GCF

Example 4 Continued

Factors of 18: 1, 2, 3, 6, 9, 18

Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

Find the common factors of 18 and 24.


The greatest possible number of milk cartons in each row is 6. Find the number of rows of each type of milk when the cook puts the greatest number of cartons in each row.

Factors; GCF

18 chocolate milk cartons6 containers per row

= 3 rows

24 regular milk cartons

6 containers per row= 4 rows

When the greatest possible number of types of milk is in each row, there are 7 rows in total.

Example 4 Continued

Factors; GCF

Factors; GCF

Lesson Quiz: Part 1


1. 50

2. 84


3. 18 and 75

4. 20 and 36

22 3 7

2 52

4

3

Factors; GCF

Lesson Quiz: Part II

Find the GCF each pair of monomials.

4x

Total rows =17

9x2

7. Jackie is planting a rectangular flower bed with 40 orange flowers and 28 yellow flowers. She wants to plant them so that each row will have the same number of plants but of only one color. How many rows will Jackie need if she puts the greatest possible number of plants in each row?

6. 27x2 and 45x3y25. 12x and 28x3

The GCF of 40 and 28 is... 4 40 ÷ 4 = 10 + 28 ÷ 4 = 7

Factors; GCF

Factor by finding the GCF

Factorizations of 12

1 12 2 6 3 4 1 4 32 2 3

The circled factorization is the prime factorization because all the factors are prime numbers. The prime factors can be written in any order, and except for changes in the order, there is only one way to write the prime factorization of a number.

Greatest Common

Factor

4. An air conditioner which cost $360 sold for $288. What was the percentage loss for this sale?

March 18, 2014

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