Algebra Chapter 5
FACTORING AND QUADRATIC EQUATIONS
Definition: A prime number is a natural number greater than 1 whose only factors are 1 and itself.
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.
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.
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.
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.
Square of a Sum: (x + y)2 = x2 + 2xy + y2
Square of a Difference: (x - y)2 = x2 - 2xy + y2
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)
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.
Every quadratic equation ax2
+ bx + c = 0 can be solved by the quadratic formula which states
If a and b are the lengths of the two legs of a right triangle whose hypotenuse
has length c,
then c2 = a2 + b2