7.1 Introduction to Factoring

Objectives: •To find the GCF of 2 or more integers•To find the GCF of two or more monomials•To find the missing factor, given a monomial and one of its factors.

Frameworks: 10.P.5

How could you rearrange 12 tiles to form a rectangle?

Factoring

To factor an integer means to express it as the product of 2 or more integers.

12 = 1 x 1212 = 2 x 612 = 3 x 4

Factoring

12 = 2 x 6 could also be expressed as12 = 2 x 3 x 2

In the second case, none of the factors can be factored further without using the factor 1.

2 and 3 are called prime numbers

Prime Numbers

A prime number is an integer greater than 1 whose only positive factors are 1 and itself.

Can you think of examples?1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31

Composite Numbers

A composite number is a positive integer that has two or more positive factors other than 1

Can you think of examples?4, 6, 8, 9, 10, 12, 14, 15

1 is neither prime nor composite

Prime factorization

A factorization where all the factors are prime numbers

12 = 2 x 2 x 336 = 2 x 2 x 3 x 3

Give the prime factorization of 90:

Choose 2 positive factors of 90. Continue factoring until all the factors are prime numbers.

Give the prime factorization of 100:

Choose 2 positive factors of 100. Continue factoring until all the factors are prime numbers.

Can there be more than 1

prime factorization of a number?

No.

GCF – Greatest Common Factor

The greatest common factor (GCF) of two or more integers is the largest integer that is a factor of all the integers.

Find the GCF of 72 and 84

72 = 12 * 6 =84 = 14 * 6 =

Find the GCF of 36 and 90

Find the GCF of 12x4 and -28x3 and 120 x2

Find the GCF of 84y3 and -96y6 and -196y5

To find a missing factor, divide.

For example, x3 * ____ = x7

x7

x3

So, the answer is x4

Find the missing factor:

a5 * ____ = a9

___ * 5a = -30a10

3a3b5 * ____ = 24a4b7

Do p. 246 1-21

FOIL Review

(x + 3)(x + 5)

First terms: x and x x2

Outside terms: x and 5 5xInside terms: 3 and x 3xLast terms: 3 and 5 15Answer: x2 + 8x + 15

FOIL pattern . . .

Multiply this: Get this: a + b a * b

(x + 1)*(x + 2) x2 + 3x + 2 3 2

(x + 3)*(x + 4) x2 + 7x + 12 7 12

(x + a)(x + b) x2 + (a+b)x + a*b

(x + 3)(x + 7)

(x + 4)(x + 5)

(x + 2)(x + 7)

Homework