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In studying mathematics we attempt to go from what we know to what we want to know, or demonstrate. We use algebra as a problem solving process for dealing with unknowns. There are some basic rules of algebra which must be learned and followed in order to have a useful system for problem solving. For each of these rules we will look at the rule, a concrete example, and an abstract example. This should make solving problems more sensible.
SOME ALGEBRA ESSENTIALS
Other sections on this page: Below on this page you can find information on Signed Numbers Distributive Property
The following notes assume that you have some basic knowledge of equations and can work with fractions and decimals as well as positive and negative numbers, but I have attempted to explain each rule as it is used.
Order of Operations
Someone once gave the memory device: Please Excuse My Dear Aunt Sally for remembering the order of operations.
P: parentheses
E: exponents
M: multiplication
D: division
A: addition
S: subtraction
remember that multiplication and division are algebraically (inversely) related:
for instance, dividing by 4 is the same as multiplying by 1/4 or 0.25
Also, think of fractions as "divided by" problems so that "8 over 4" means 8 ÷ 4 = 2
As stated above the rule is often remembered as:
multiply (or divide) before you add or subtract and works like this example 3 * 7 + 3 * 5 = 21 + 15 = 36
This oversimplified "Please Excuse My Dear Aunt Sally" rule is not exactly correct. This much is good: grouping is the first priority and exponents (with their base) represent a single term . After that, however, the accepted procedure is to multiply and divide from left to right as you come to it, then add or subtract from left to right as you come to it. Since multiplication and division are inverse processes, then there is no priority given to either. And, since addition and subtraction are inversely related, then neither gets priority over the other.
I prefer to use:
P
E
M & D
A & S
Let's look at example: 
In this sequence the interior section
helps illustrate the proper order.
Beginning on the left,
so the problems becomes 
caution: though order doesn’t matter for addition and multiplication, it does matter for division and subtraction
but
and
but
Multiplication and addition are "commutative" but subtraction and division are not "commutative."
but
(fractions are division expressions) and
but
remember that some operations result in no change in value. These are called identities. the identity for multiplication and division is 1 and the identity for addition and subtraction is 0.
4 * 1 = 4 and 4 + 0 = 4
remember that powers are multiplication of a base by itself
and
42 = 4 * 4 = 16 and 43 = 4 * 4 * 4 = 64
because of notation, you must compute the powers before you multiply
3 * 42 = 3 * 4 * 4 = 48 and 3x2 = 3 * x * x (the 3 is not squared)
when you have a quantity raised to a power, then everything in the parentheses is the base
concretely: (3+7)3 = 10 * 10 * 10 = 1000
do NOT distribute the power to the individual parts of the base!
(3+7)3 = 10 * 10 * 10 = 1000 ¹ 33 + 73 = 27 + 343 = 370
remember that the sum of two opposites is 0 (the identity for addition is 0)
-7 + 7 = 0 is the same as 7 - 7 = 0 and 21 + -21 = 0 is the same as 21 - 21 = 0
When adding numbers, work from left to right and honor the signs.
The sign of a number indicates its direction from 0 on a number line. The magnitude of
the number is found by its absolute value. The absolute value of a number indicates its
distance from zero without regard to its direction.
and 
remember that the product of a positive and a negative is negative and the product of two negatives is positive
-7 * -6 = 42
-7 * 6 = -42 and 7 * -6 = -42
because this is the same as 
and
because this is the same as 
and the –1*-1 =1
We use these rules for the expressions abstractly until we substitute specific values.
If x is given a positive value and y a negative value then the final
product is negative 
If x is given a negative value and y a positive value then the final
product is negative 
If x is given a negative value and y a negative value then the final
product is positive 
when you see 
think of it as the opposite of
which means the same as
so, 
because 4 is the base and
because –4 is the base
think of the term as: base and exponent which must stay together, and then any multiplier as the coefficient of that term
This means that numbers raised to an even power will give a positive product.
And, it also means that a negative base raised to an odd power will give a negative
product. 
remember that powers only apply to their own base
-x2 = -( x * x) because the - sign stands for a coefficient of -1, so -x2 = -1 * x * x
now we can see that:
-72 = -(7 * 7) = -49 because 7 is the base and (-7)2 = 49 because -7 is the base
Distributive Property and factoring
this property is the backbone to algebraic operations
You can demonstrate how this must be true by some simple, concrete examples.
36 = 3 * 12 = 3(7+5) = 21 + 15 = 36 or
36 = 4 * 9 = (3 + 1)9 = 3 * 9 + 1 * 9 = 27 + 9 = 36
48 = 4 * 12 = 4(20 - 8) = 80 - 32 = 48 or
48 = (15 - 7)6 = 15 * 6 – 7 * 6 = 90 - 42 = 48
so we can see that 
and 
since multiplication is commutative, then this is also true: 
this allows us to take another step:
the distributive property also demonstrates how we can find common factors from terms and "factor" them out
is rewritten as 
when we factor out a 3 from
each term. 3 is called the common factor.
and so we can see that 
when
we factor out the common factor 4
remember that powers only apply to their own base
you can see that there are actually three
2’s which are factored out so that the common factor was 23 = 8 . This
gives us 
It is important that we be able to distribute to find the product of binomials
also
There is a memory tool known as "FOIL" that helps to multiply two binomials:
It is First Outside Inside Last
Multiply the two first terms, then the two outside terms, then the two inside terms, then the two last terms.
Using the above example
There are three special cases for the multiplication of two binomials:
You should memorize them:
 |
this product is known as a perfect square trinomial
this product is known as a perfect square trinomial
this product is known as the difference between two squares |
Again, you should memorize these for use in both distributing and in factoring
simple factoring problems quickly become easy to do in our heads, but practicing will help us with more abstract problems
for instance, you can probably do the product of these two binomials and get
which looks a lot more messy than the original
so, it makes sense that when we see 
that we
would want to be able clean it up some. Now we know that we can factor
by identifying x - y as the common
factor and factoring it out to end up with 
remember that powers only apply to their own base
in 
you must not distribute the factor (x+7)
into the quantity that is being raised to the power of 2. (x+7)
is the coefficient and (x-3)
is the base
we should note that we could have gotten the same answer by working in a different order
this is an illustration of the commutative property for multiplication
how to have
