Elementary Algebra Chapter 5
FACTORING AND QUADRATIC EQUATIONS

From Section 5.1
Definition: A prime number is a natural number greater than 1 whose only factors are 1 and itself.

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: 3xy² + 6x²y³ may be written as 3(xy² + 2x²y³)
                      3xy² + 6x²y³ may be written as x(3y² + 2xy³)
                      3xy² + 6x²y³ may be written as 3x(y² + 2xy³)
                      3xy² + 6x²y³ may be written as 3xy(y + 2xy²)
                      3xy² + 6x²y³ may be written as 3xy²(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 x² +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 ax² +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
Square of a Sum:                      (x + y)² = x² + 2xy + y²

Square of a Difference:              (x -  y)² = x² -  2xy + y²

Difference of Two Squares:     x² -  y² = (x - y)(x + y)

Difference of Two Cubes:         x³ -  y³ = (x - y)(x² + xy + y²) 

Sum of Two Cubes:                   x³ + y³ = (x + y)(x² - xy + y²) 

From Section 5.5

FACT: If a and b are real numbers and ab = 0 then a = 0 or b = 0.

FACT: If the quadratic equation ax² + 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 ax² + 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 c² = a² + b²