**Elementary
Algebra Chapter 5
FACTORING AND QUADRATIC EQUATIONS**

**From
Section 5.1
Definition: **A

**FACT:**
The first few prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37,
41, 43, 47

The first **1000
primes **

**Definition:
**A
number is in **prime-factored form**
if it is written as the product of prime numbers.

**Definition:
**If
number is in prime-factored form the product of prime numbers is called the
**prime factorization**
of the number.

**WEB
Practice:**
Allows you to **perform
factorizations **using the factor tree

**Example:**
When we write 12 = (2)(2)(3), then 12 is said to be in prime-factored form and
(2)(2)(3) is the prime factorization of 12.

**Definition:
**The
** Greatest Common Factor (GCF)**
of two integers is the largest natural number which will divide both integers
evenly.

**Definition:
**The
** Greatest Common Factor (GCF)**
of two monomials is the product of all numbers and letters which divide each
of the monomials.

**FACT:**
A sum (or difference) of terms may be written as a product by factoring out
any factors common to each term.

**Example:**
3xy^{2} + 6x^{2}y^{3 }may be written as 3(xy^{2}
+ 2x^{2}y^{3})

3xy^{2}
+ 6x^{2}y^{3 }may be written as x(3y^{2} + 2xy^{3})

3xy^{2}
+ 6x^{2}y^{3 }may be written as 3x(y^{2} + 2xy^{3})

3xy^{2}
+ 6x^{2}y^{3 }may be written as 3xy(y + 2xy^{2})

3xy^{2}
+ 6x^{2}y^{3 }may be written as 3xy^{2}(1 + 2xy) in
this case the GCF has been factored out.

**FACT:**
Factoring a term out of a sum (or difference) of terms is reversing a multiplication
done with the distributive property.

**From
Section 5.2**

**PROCESS:
**To
factor a trinomial of the form x^{2} +bx + c into a product of two linear
factors of the form x + k we proceed as follows:

1) Find
two integers h and k whose product is c and whose sum is b.

2) Write
the factorization using the appropriate choice of signs so the product is equal
to the trinomial.

**From
Section 5.3**

**PROCESS:
**To
factor a trinomial of the form ax^{2} +bx + c into a product of two
linear factors of the form px + k we proceed as follows:

**The
following elaborate example illustrates some of the difficulties with factoring
second degree polynomials.**

**From
Section 5.4
**

**Difference
of Two Squares: **** x ^{2}
- y^{2} = (x - y)(x + y)**

**From
Section 5.5
**

**FACT:**
If the quadratic equation
ax^{2} + bx + c = 0 can be factored into linear factors (mx + k)(nx
+ p) then the quadratic equation can be solved by solving the two linear equations
mx + k = 0 and nx + p = 0.

**Quadratic
Formula:**
Every quadratic equation ax^{2}
+ bx + c = 0 can be solved by the quadratic formula which states

**Pythagorean
Theorem:**
If a and b are the lengths of the two legs of a right triangle whose hypotenuse
has length c,

then c^{2} = a^{2} + b^{2}