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Prerequisites: Fundamental Concepts of Algebra > Study Guide
Section P.1
Hints:
- 1. Natural numbers are the numbers used for counting.
Think about counting a set of objects. Where would you begin? You
would begin at 1. Therefore, the set of natural
numbers begins with 1, not 0.
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2. Remember! Positive
numbers are to the right of zero on a number line while negative numbers are
to the left of zero on a number line. Therefore, numbers
get larger as you travel from left to right on the number line.
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- 3. Absolute value is the distance that a number is from zero
on the number line, and a distance cannot be negative. Therefore,
the answer coming out of absolute value symbols (| |) is always positive,
no matter what is inside the symbols. Any negative sign in front of absolute
value is applied after the absolute value symbols are removed.
Example: |3| = 3 and |-3| = 3
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-|3| = -3 and -|-3| = -3 (Since negative sign is outside of | |.)
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- 4. The Order of Operations can get tricky when a problem has
both addition and subtraction or both multiplication and division. Addition
and subtraction are on equal levels so do whichever one comes first, as you
look at the problem from left to right. This holds true for multiplication and
division as well. Of course, parentheses can change the order. Always
do the operations in parentheses first.
5. The commutative properties have to do with the
order of the numbers, i.e., the numbers move. Think of commutative as
"commute" or moving. Note: There is no "n" in this name, it
is not "communtative".
The associative properties have to do with changing
the grouping of the numbers. This can be remembered by thinking
of an "association" as a group. Parentheses define the grouping.
The distributive property "distributes" or shares
(by multiplying) a value among a group in parentheses, like we might
share a bag of candy with a group of friends, giving a piece to each one.
- 6. Variables can be any single letter, upper or lower case.
Traditionally, x and y are used frequently in algebra.
However, feel free to use other variables, especially if they help you
remember the value for which they stand; like t
for time or w for width,
etc.
- 7. An algebraic expression is made up of one or more terms,
separated by addition or subtraction. Each term has a number in front. This
is the numerical coefficient and should
include the positive or negative sign. If there is no variable, the term is
called a constant. The terms in an expression can really be in any
order. However, traditionally, the
terms are put in alphabetical order with exponents decreasing for terms with
the same variables. Constants are usually at the end of an
expression. Although order does not affect the value of the expression,
following tradition may make some problems easier to work.
8. An expression is simplified (not solved)
and usually does not result in a numeric answer. Evaluate an algebraic
expression by first substituting the given value(s) for the variables in the
expression, then follow the order of operations. Just be sure to be
careful to apply them correctly when variables are involved.
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- 9. Be careful to add or subtract only like (similar) terms. Like
terms have exactly the same variable(s) and matching exponents. The
coefficients do not have to match in similar terms. An
expression is simplified when all the parentheses have been correctly
removed, and like terms are combined.
Section P.2
Hints:
- 1. The properties of exponents apply to numbers with the same base and any
form of exponent (integer, fractions, decimals, irrational numbers, square
roots). The table on page 18 explains the properties. The property
that students seem to have the most trouble with may be the quotient rule.
This is particularly true when there is a negative exponent in the
denominator. Remember! Subtract the
exponent in the denominator from the exponent in the numerator when dividing
like bases.
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- For example:
.
- 2. Anything raised to the zero power equals one, no
matter how complex the base is.
- 3. It may be helpful to think of the negative sign
in a negative exponent as "one over".
So
is "one over" 
or 
.
- 4. Scientific notation allows us to easily work with very large or very
small numbers by converting them to numbers multiplied by a power of ten. Remember
that a positive exponent for 10 results in a large number (ex.
)
and a negative exponent for 10 results in a small
number (ex. 
).
When doing arithmetic with scientific notation, just follow the properties
of exponents for the powers of 10 and regular arithmetic for the leading
numbers. Convert any final answers to scientific notation.
- 5. A negative exponent of ten is a small number. A negative sign before
the leading number in scientific notation makes the whole number negative.
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Section P.3
Hints:
- 1. When simplifying square roots, it is very helpful to know
many of the perfect squares and their square
roots.
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- 2. To simply a square root, follow these steps:
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1) Factor the number under the radical into two
factors so that are one of the factors is a perfect square (like 4, 9, 16, etc.).
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2) Take the square root of the perfect
square.
Its square root will sit outside the radical (in the front).
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3) Leave the remaining factor underneath the radical.
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- Example:
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3. Donít let arithmetic with square roots confuse you. In general,
follow the procedure below.
Adding and Subtracting: 1. Simplify. 2.
Perform operations.
Example: 
Multiplying and Dividing: 1. Perform operations.
2. Simplify.
Example: 
4. To rationalize a denominator containing a radical, look at the number
under the radical sign in the denominator. Find the smallest number you
can use to multiply that number by to turn it into a perfect square. Then
multiply by the square root of this number in the numerator and denominator of
the fraction (like multiplying by one). Then, if necessary, simplify and
reduce.
Example: 
if you multiply the 12 by 3 you
get 36, a perfect square.
Or just multiply by the denominator
over itself and simplify. This method may require a bit more work.
5. When considering a rational exponent, sometimes it is helpful to use
the formula 
.
In this formula "p" means power
and "r" means root. A
rational exponent is always the power divided by the root or "power over
root". Remembering this will help you convert rational
exponents to radicals and radicals to rational exponents.
Section P.4
Hints:
- 1. Polynomials are types of
algebraic expressions. Therefore, Hints 7, 8,
and 9 from P.1
apply.
-
- 2.
When subtracting polynomials,
it is very important to change the sign of each term
inside the parentheses that comes after the subtraction symbol.
-
-
- Ex.
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- 3.
When multiplying two binomials together remember that you can
use the FOIL Method which is explained in
depth on pages 40 and 41. You should remember that the FOIL
METHOD is using the distributive property.
Each term in the first binomial is distributed over each term in the second
binomial. If you know how to use the distributive property, then you can
use the FOIL METHOD.
4. Remember! When you square
a binomial you will never get just two terms. There will always be three
terms.
Ex. 
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Section P.5
Hints:
1. The
opposite of multiplying two binomials is to
factor or break down a polynomial expression. Several methods for
factoring are given in the text. Be persistent in factoring!
It
is normal to try several pairs of factors, looking for the right ones.
The more you work with factoring, the easier it will be to find the correct
factors. Also, if you check your work by using the FOIL method, it is virtually
impossible to get a factoring problem wrong.
- 2. Remember!
When factoring, always take out any factor that is
common to all the terms first.
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- 3.
When factoring a trinomial, here are some hints to help you decide about the
signs.
1. If the last term of the trinomial is
positive, then the signs in
the middle of the binomial factors will the same. You will
use
the sign of the second term in
the trinomial as that sign.
Ex. 
2. If the last
term of the trinomial is negative, then the signs in the middle of the binomial factors will be different from each other. In other words,
one will be positive and the other will be negative.
Ex. 
4. There is a very helpful chart on
page 55 that gives the steps of a strategy for factoring.
Section P.6
Hints:
- 1. Remember! When you are looking for the domain of a rational expression, you are
looking for values that the variable cannot be equal
to. Follow these steps when looking for the domain of a
rational expression.
- 1. Factor the denominator, if necessary.
- 2. Set each factor not equal to
zero and solve.
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- 3. Solve the equations from Step 2.
- Ex.
-
- 2. Before multiplying or
dividing rational expressions, you should always factor each of the numerators and denominators
of the expressions.
- 3. After factoring the numerators and denominators of the
rational expressions you should divide by any common
factors. Think of dividing by a common factor as "canceling out" these factors. You may only "cancel" numerators with
denominators. Never cancel a numerator with another numerator
or a denominator with another denominator.
Ex.
4. When dividing rational
expressions, remember to get the reciprocal of the
divisor (the expression that you are dividing by) and change the division into multiplication.
5. When adding or subtracting rational expressions, they must
have a common denominator. Finding the lowest common denominator
(LCD) is the first step when adding unlike
expressions. The steps for finding the LCD are on
page 64.
6. When adding or subtracting
rational expressions, remember that you are still adding or subtracting algebraic
expressions. All the rules from P.4 apply,
so watch your signs when subtracting.
Copyright ©
2003
by Prentice Hall, Inc.
A Pearson Company Legal Notice
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