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= + ÷ ± ∆ ∞ × π Σ f(x) % ≤ ! ≠ * |
Mathematics attempts 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.
The following notes assume that you have some basic knowledge of equations and can work with fractions and decimals, positive and negative numbers.
Order of Operations
remember that multiplication and division are algebraically related:
for instance, dividing by 4 is the same as multiplying by 1/4 or 0.25
multiply (or divide) before you add or subtract
3 * 7 + 3 * 5 = 21 + 15 = 36
caution: though order doesn’t matter for addition and multiplication, it does matter for division and subtraction
but 
and 
but 
but 
and 
but 
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)
think of the term as two parts, base and exponent, and any multiplier as the coefficient of that term
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
Signed Numbers
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 which is defined as the number's distance from 0
without regard to its direction.
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
This means that numbers raised to an even power will give a positive product.
But, 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 = 27 + 9 = 36
48 = 4 * 12 = 4(20 - 8) = 80 - 32 = 48 or
48 = 8 * 6 =(15 - 7)6 = 90 - 42 = 48
so we can see that 
and 
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 also important that we be able to distribute and factor with the products of binomials
also
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