Polynomials are continuous and smooth everywhere.

- A continuous function means that it can be drawn
**without picking up your pencil**. There are no jumps or holes in the graph of a polynomial function. - A smooth curve means that there are
**no sharp turns**(like an absolute value) in the graph of the function. - The
**y-intercept**of the polynomial is**the constant**term a_{0}.

- If the leading coefficient, a
_{n}, of the polynomial is**positive**, then the right hand side of the graph will**rise**towards + infinity. - If the leading coefficient, a
_{n}, of the polynomial is**negative**, then the right hand side of the graph will**fall**towards - infinity.

- If the degree, n, of the polynomial is
**even**, the left hand side will do the**same**as the right hand side. - If the degree, n, of the polynomial is
**odd**, the left hand side will do the**opposite**of the right hand side.

Get used to this even-same, odd-changes notion. We will be seeing it a lot ("lot" is a mathematical term meaning you will be sick of it - but that it's probably something that's really important.)

- An nth degree polynomial in one variable has
**at most n real zeros**. There are exactly n real or complex zeros (see the Fundamental Theorem of Algebra in the next section). - An nth degree polynomial in one variable has
**at most n-1 relative extrema**(relative maximums or relative minimums). Since a relative extremum is a turn in the graph, you could also say there are at most n-1 turns in the graph.

If f is a polynomial function in one variable, then the following statements are equivalent

- x=a is a
*zero*or*root*of the function f. - x=a is a
*solution*of the equation f(x)=0. - (x-a) is a
*factor*of the function f. - (a,0) is an
*x-intercept*of the graph of f.

The claim is made that there are at most n real zeros. There is no claim made
that they are all unique (different). Some of the roots may be repeated. These
are called **repeated roots**. Repeated roots are tied to a concept
called multiplicity. The **multiplicity** of a root is the number
of times a root is an answer. The easiest way to determine the multiplicity
of a root is to look at the exponent on the corresponding factor.

Consider the following

f(x) = (x-3)^2 (x+5) (x+2)^4

The roots to the function will be x=3 with multiplicity 2, x=-5, and x=-2 with multiplicity 4. It is assumed, and therefore unnecessary to write, a multiplicity of 1.

The multiplicity of a root, and likewise the exponent on the factor, can be used to determine the behavior of the graph at that zero.

- If the multiplicity is
**odd**, the graph will**cross**the x-axis at that zero. That is, it will change sides, or be on opposite sides of the x-axis. - If the multiplicity is
**even**, the graph will**touch**the x-axis at that zero. That is, it will stay on the same side of the axis.

Wait - it seems I've mentioned that before. I feel like Tweety-Bird when I thot I thaw a puddy tat. I did, I did. Odd changes, even stays the same. I call it OCES. Get used to it - it will be a recurring theme.

Odd Changes, Even Same

Here are some places you will be using the concept of Odd Changes, Even stays the Same

- The left hand behavior of a polynomial function.
- If the degree of the polynomial is
*Odd*, the left hand*Changes*from the right hand - If the degree of the polynomial is
*Even*, the left hand does the*Same*as the right hand

- If the degree of the polynomial is
- The behavior of a polynomial function at the x-intercepts
- If the multiplicity is
*Odd*, the graph will*Change*sides and cross the axis - If the multiplicity is
*Even*, the graph will stay on the*Same*side and just touch the axis

- If the multiplicity is
- Determining the solution to inequalities (this is the key to finding answers
really quickly)
- If the multiplicity is
*Odd*, the sign will*Change*at the critical number - If the multiplicity is
*Even*, the sign will stay the*Same*at the critical number

- If the multiplicity is
- The behavior of a rational functions (later in the chapter) the x-intercepts
- If the multiplicity is
*Odd*, the graph will*Change*sides and cross the axis - If the multiplicity is
*Even*, the graph will stay on the*Same*side and just touch the axis

- If the multiplicity is
- Vertical asymptotes of rational functions (later in the chapter)
- If the multiplicity is
*Odd*, the graph will*Change*sides and one side of the vertical asymptote will rise to positive infinity while the other side falls to negative infinity. - If the multiplicity is
*Even*, the graph will stay on the*Same*side, and both sides of the vertical asymptote will rise to positive infinity or both sides will fall to negative infinity.

- If the multiplicity is
- Determining the sign of the cofactor of an element of a matrix (chapter
6)
- If the sum of the row and column the element is in is
*Odd*, the cofactor will*Change*and be the opposite of the minor. - If the sum of the row and column the element is in is
*Even*, the cofactor will be the*Same*as the minor.

- If the sum of the row and column the element is in is

Polynomials are continuous functions which mean that you can't pick up your pencil while graphing them.

- Question:
- If at some point, you're below the x-axis, and at another point you're above the x-axis, and you didn't pick up your pencil while moving from one point to the other - what happened?
- Answer:
- You crossed the x-axis, had a zero or root of the function, found a solution, etc.

Now, take that concept a little bit farther. Take any two y-values. If they're not the same, then you had to hit every y-value between the two when moving from one to the other. The Intermediate Value Theorem states that formally.

What it's primarily used for, however, is to find the zeros of a continuous function.