College Algebra and Probability

We will learn to:
Solve linear and quadratic equations and inequalities
Graph lines, circles, and parabolas
Graph inequalities
Apply linear , quadratic, and other models to solve problems
Use exponential and logarithmic functions
Count

You will need

the book
Jerome E. Kaufmann, College Algebra, 6/e

a calculator
At minimum, your calculator should have
arbitrary exponents or
exponentials
logarithms
factorials
permutations
combinations

Weekly homework — 15%

Problems listed in syllabus
Each week’s problems due the following Wednesday
6–8 problems graded

Three hour exams — 15% each = 45%

Final exam — 40%

Attend class

Not mandatory
No attendance points or participation points
A good deal of lecture material will not be in the slides

Do not make yourself a distraction

No loud conversations
No newspaper rustling
No stinky or noisy food

Most importantly, ask questions!

  Review of Basic Concepts
The Real Numbers
Fractions

  Integer Exponents
Positive Integer Exponents
Zero as an Exponent
Negative Integer Exponents
Scientific Notation

  Radicals
Definition and Notation
Simplest Radical Form

  Fractional Exponents

The rationals are the set of all possible quotients formed using
integers
A rational number can be written as where p and q are
integers with

A rational number can also be written in decimal form, and will
either terminate

or repeat

Irrational numbers cannot be expressed as fractions. This means
they have decimal representations that neither terminate nor
repeat.



The reals are difficult to define. For now, let’s just say the reals
comprise the rationals and the irrationals.

We can simplify fractions by canceling the same thing from the
numerator and the denominator.

To add or subtract fractions , get a common denominator, and then
add or subtract the numerators.

To multiply fractions , multiply the numerators and denominators
separately. Cancel first if possible.

To divide fractions, multiply by the reciprocal of the divisor .

The base is repeatedly multiplied by itself
The exponent indicates the number of factors

Zero exponents should satisfy the same properties as positive
exponents. Let m = 0 in Property 1:

So, unless b is zero, b⁰ must be 1.

Again, we want negative exponents to satisfy the same properties
as positive exponents. Set m = −n in Property 1:

So b^-n must be the reciprocal of bⁿ.

To write a number in scientific notation,

  Move the decimal point to the immediate right of the first
nonzero number—this produces n
  Count the number of places the decimal point moved—this
gives you |k|
  If you moved the decimal point to the left, k is positive; if you
moved it right, k is negative.

Evaluate
Evaluate
  Simplify . Use only nonnegative exponents.
Express 0.000000078 in scientific notation.
Use scientific notation to evaluate

Definition of

The book’s definition is wrong!

Note: Even though is 2, not −2.

For any real number b, if exists, it is nonnegative.

Definition of

Definition For any positive integer n and nonnegative real number a, if and only if If n is odd, this holds for any (possibly negative) real number a.

Examples

Properties of Radicals If and are real numbers, then

Notice that there is no property for A sum of radicals
generally cannot be simplified.

Also notice how similar these properties are to the properties of
exponents. .

Definition An expression is said to be in simplest radical form if: No fraction appears within a radical sign . No radical appears in the denominator. No radicand contains a factor that is a perfect power of the index.

An example of the last condition is is a factor of 75, and
25 is a perfect square .

To clear a radical from the denominator, multiply the numerator and
the denominator by a radical that will allow you to remove the
radical sign.

In case there is a sum ( or difference ) of radicals in the
denominator, say multiply by the conjugate

  Express in simplest radical form.

  Express in simplest radical form.

  Rationalize the denominator and simplify


  Rationalize the denominator and simplify

Recall that . Suppose m = 1/n. Then



So   is the nth root of b.

Definition For any positive integer n and any real number b, if exists, then

Roots are Exponents!
Definition of

Because we have



and
 

Definition For any positive integer n and any real number b, if m/n is a rational number written in lowest terms, and if exists, then

  Evaluate

  Evaluate

  Evaluate

  Express in simplest radical form.

  Simplify into one radical.