MULTIPLE ROOTS
POINT OF INFLECTION
WHEN WE STUDIED quadratic equations, we saw what it means for a polynomial to have a double root.
We now want to learn about the graphs of polynomials that have multiple roots.
For example, this polynomial
y = (x 1) = (x
LOGARITHMIC AND EXPONENTIAL
FUNCTIONS
The function is defined only for positive values of x.
logb(4), for example, makes no sense. Since b is always positive, no power of b can produce a
negative number.
The range of the function is all real numbers.
Th
FACTORIALS
BY THE SYMBOL n! ("n factorial") we mean the product of consecutive numbers 1 through n.
n! = 1 2 3 (n 2)(n 1)n
= n(n 1)(n 2) 3 2 1
The order of the factors does not matter, whether backwards or forwards.
0! is defined as 1.
0! = 1
(The usefuln
The Formal Rules of Algebra
ALGEBRA is a method of written calculations. And a calculation is replacing one set of symbols with
another. In arithmetic we may replace the symbols '2 + 2' with the symbol '4.' In algebra we may replace
'a + (a)' with '0.'
a
RATIONAL AND IRRATIONAL NUMBERS
The following numbers of arithmetic are the counting-numbers or, as they are called, the natural numbers:
1, 2, 3, 4, and so on.
(At any rate, those are their numerals.)
If we include 0, we have the whole numbers:
0, 1, 2,
FUNCTIONS
What is a function?
WHEN ONE THING DEPENDS on another, as for example the area of a circle depends on the radius - in
the sense that when the radius changes, the area also will change - then we say that the first is a "function"
of the other. Th
INTRODUCTION TO GRAPHS
WE WILL BEGIN with some basic vocabulary.
First, a cordinate. A cordinate is a number. It labels a point on a line.
The cordinates 0, 1, 2, 3, etc. label those points. They are the "addresses" of those points.
A cordinate axis is a
THE VOCABULARY OF
POLYNOMIAL FUNCTIONS
1. When numbers are added or subtracted, they are called terms. This -4x2 + 7x 8
- is a sum of three terms. (In algebra we speak of a "sum," even though a term may be subtracted.)
When numbers are multiplied, they ar
THE ROOTS, OR ZEROS,
OF A POLYNOMIAL
1. What is a polynomial equation?
It is a polynomial set equal to 0. P(x) = 0.
Example. P(x) = 5x 4x + 7x 8 = 0
2. What do we mean by a root, or zero, of a polynomial?
It is a solution to the polynomial equation, P(x)
THE SLOPE
OF A STRAIGHT LINE
IN LESSON 33 OF ALGEBRA we discuss the equation of a straight line. Sketching the graph of the
equation of a line should be a basic skill.
Consider this straight line.
The (x, y) cordinates at B have changed from the cordinate