**The Rules of Algebra**

Algebra is arithmetic with numbers and letters. Arithmetic consists of the following operations

Addition Subtraction

Multiplication Division

Powers Roots and Logarithms

Addition is repeated counting. Subtraction reverses addition. Multiplication is repeated addition. Division reverses multiplication. Powers or exponents are repeated multiplication. Roots and logarithms reverse powers.

To apply an operation to two letters, write the letters down with the operation between them,

*a* + *b a
– b*

* ab a*/*b*

* a ^{b} * or log

The answer to an arithmetic problem with letters is an expression. We can also do arithmetic with expressions. Copy down the expressions in parentheses and put the operation between the sets of parentheses.

(2*x*) + (*a*
+ *b*) (2*x*) - (*a*
+ *b*)

(2*x*)(*a* + *b*) (2*x*)/(*a* + *b*)
or

(*a* + *b*)^{2}* ^{x}* or log

A fraction is the answer to a division problem. If you use the vertical format, the fraction bar acts like parentheses.

**The Order of
Operations**

Combine
whatÕs inside parentheses.

Exponents
and roots

Multiplications
and divisions

Additions
and subtractions.

**Rules for
Parentheses**

*a* + *b* = *b* + *a ab* = *ba* Commutative
rules

*a* + (*b* +
*c*) = (*a* + *b*) +
*c a*(*bc*) =
(*ab*)*c* Associative
rules

0
+ *a* = *a* 1*a* = *a* Identity
rules

-*a* + *a* =
0 *a* = 1 Inverse
rules (*a* 0)

*a*(*b*
+ *c*) = *ab* + *ac* Distributive
rule

**Rules
for Exponents**

** Definition** *a ^{n}*
=

** ***n* factors (*n* = 1, 2, 3, . . . )

*a ^{n}a^{m}
*=

(*a ^{n}*)

*a*^{0} = 1

**Definitions**

**Terms** are added and subtracted

**Factors** are multiplied and divided

There are two
kids of factors, number factors and letter factors. There need only be one
numerical factor in a term, because the commutative and associative rules
enable you to move all of your numerical factors together and multiply them up
to get a single numerical factor, which the commutative rule says you can write
at the left hand side of the term. This numerical factor is called the **coefficient**. The letter factors are called **variables**. If your term contains several factors of the same
variable, you can tell your reader how many factors of that variable there are
by using a **power** or an **exponent**. The number of variable factors in a term is called
the **degree** of the term.

An expression
which is made up of only addition, subtraction, and multiplication is called a **polynomial**. The coefficients in a polynomial can be fractions,
but there are no variables in denominators. The degree of a polynomial is the
degree of the highest degree term. Polynomials of degree one are called **linear**. Polynomials of degree two are called **quadratic**. Polynomials of degree three are called **cubic**. Polynomials of degree four are called **quartic**. Polynomials of degree five are called **quintic**. A polynomial with one term is called a **monomial**. A polynomial with two terms is called a **binomial**. A polynomial with three terms is called a **trinomial .**

**Like terms** are terms with exactly the same variable factors.
The distributive property enables you to combine like terms. If the two terms have exactly the same
variable factors, you can factor them all out leaving nothing but numbers
inside parentheses, and whenever you have nothing but numbers inside
parentheses, you can combine them into a single numbers. To **combine
like terms**, add or subtract the
coefficients. The common variable factors give us the variables in the answer.

The distributive
property also allows us to remove parentheses. To remove parentheses, multiply
the factor outside the parentheses by all the terms inside the parentheses, and
add the products.

**Arithmetic with
Polynomials**

To add or
subtract or multiply polynomials, remove parentheses and combine like terms.
For multiplying, this amounts to multiplying each term in one polynomial by
each term in the other. To multiply terms, multiply the coefficients and add
the exponents on each variable. The number of terms in the product will be
equal to the product of the number of terms. Of course, there may well be like
terms which you will need to combine.

**Long Division of
Polynomials**

** **

**Definitions**

The polynomial you are dividing
by is called the **divisor.**

The polynomial you are dividing
it into is called the **dividend**.

The answer is called the **quotient**.

The difference between the
quotient times the divisor and the dividend is called the **remainder**.

1.
Divide the first term in the dividend into the first term in
the divisor. This gives you the first term in the quotient.

2.
Multiply this term in the quotient by the divisor and subtract
this product from the dividend.

3.
Repeat the process with the remainder until you have a
remainder whose degree is smaller than the degree of the divisor.

**Fractions**

Definition: A **fraction** (or rational expression) is the answer to a
division problem of polynomials.

**The Fundamental
Fact of Fractions**

If you multiply
(or divide) the top and bottom of a fraction by the same thing, you get a
different name for the same number.

**Reducing or
Simplifying Fractions**

Factor
the top and bottom until you get factors that cannot be factored further. If
you find the same factor on both the top and bottom, you can cancel them. If
after factoring the top and bottom as much as possible, if there are no common
factors in the top and bottom, the fraction is reduced to lowest terms.

**Adding
(Subtracting) Fractions**

If
you have common denominators, add (subtract) the numerators.

If
not, find common denominators.

To
find common denominators, factor all the denominators and fill in the missing
factors. You can multiply the bottom by whatever you want so long as you
multiply the top by the same thing.

**Multiplying
Fractions**

Multiply
the tops and multiply the bottoms.

You
can cancel either before or after you multiply.

**Dividing Fractions**

Invert
the divisor and multiply.

**Equations**

DefÕn: An **equation** consists of an equals sign with and expression on
either side.

DefÕn: A **solution** to an equation is something you can substitute in
for a variable in an equation, which would make the same thing come out on both
sides.

DefÓn: Two equations are equivalent if they
have the same solution(s)

The Fundamental Technique for
Equations

You
can make any change you want on one side of an equation so long as you make the
same change on the other side.

One
of the most common techniques is to get rid of a term on one side by
subtracting it from both sides. When you get rid of a term on one side, it pops
up on the other side with its sign changed. Moving a term from one side to the
other and changing its sign is called **transposing** the term.

If
you move a factor from one side to the other, move it across the fraction bar.

**Steps in solving
first degree equations**

1.
Clear Denominators: Multiply both sides by a common
denominator.

2.
Simplify: Remove parentheses and combine like terms.

3.
Transpose known terms to one side and unknown terms to the
other.

4.
Combine.

5.
Divide both sides by the coefficient of the unknown.

**Steps in solving
quadratic equations by factoring.**

- Clear Denominators: Multiply both sides by a common
denominator.
- Simplify: Remove parentheses and combine like terms.
- Transpose all terms to one side leaving a 0 on the other.
- Combine.
- Factor.
- Set the factors = 0.
- Solve the two resulting first degree equations.
- Check.

**Steps in solving
quadratic equations by completing the square.**

- Clear Denominators: Multiply both sides by a common
denominator.
- Simplify: Remove parentheses and combine like terms.
- Transpose known terms to one side and unknown terms
to the other
- Combine
- Divide both sides by the coefficient of the square
term.
- Add the square of half the coefficient of the first
degree term to both sides.
- Simplify to get a perfect square on one side and a
number on the other.
- Take square roots of both sides.
- Transpose the number with the unknown to the other
side.
- Check.

**Steps in solving
quadratic equations using the quadratic formula.**

1.
Clear denominators: Multiply both sides by a common
denominator.

2.
Simplify: Remove parentheses and combine like terms.

3.
Transpose all terms to one side leaving a 0 on the other.

4.
Combine.

5.
Substitute the coefficients into the quadratic formula.

6.
Check.

Polynomial equations of degree higher than two are beyond the scope of this discussion. There is a cubic and a quartic formula, involving radicals, to get rid of the powers, but beyond that it can be proven that solution by radicals will not always work. One reason the cubic and quartic formulas are not often studied is because they are so complicated as to be computationally useless.

**Steps in solving
rational equations.
**

After clearing denominators, you will have a polynomial equation. If it is first or second degree, the steps above wiill suffice.

**Logarithms.**

If *b ^{x}* =

log* _{b} b^{x}* =

log_{b} *xy* = log_{b} *x* + log_{b} *y*

log* _{b} x^{r}* =

change of base formula