3.2 - Polynomial Functions of Higher Degree

Graphs of Polynomials

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 a₀.

Leading Coefficient Test (right hand behavior)

• 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.

Degree of the Polynomial (left hand behavior)

• 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.)

Zeros of a Polynomial Function

• 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.

Real Zeros

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.

And the beautiful thing is ...

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

1. The left hand behavior of a polynomial function.
   1. If the degree of the polynomial is Odd, the left hand Changes from the right hand
   2. If the degree of the polynomial is Even, the left hand does the Same as the right hand
2. The behavior of a polynomial function at the x-intercepts
   1. If the multiplicity is Odd, the graph will Change sides and cross the axis
   2. If the multiplicity is Even, the graph will stay on the Same side and just touch the axis
3. Determining the solution to inequalities (this is the key to finding answers really quickly)
   1. If the multiplicity is Odd, the sign will Change at the critical number
   2. If the multiplicity is Even, the sign will stay the Same at the critical number
4. The behavior of a rational functions (later in the chapter) the x-intercepts
   1. If the multiplicity is Odd, the graph will Change sides and cross the axis
   2. If the multiplicity is Even, the graph will stay on the Same side and just touch the axis
5. Vertical asymptotes of rational functions (later in the chapter)
   1. 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.
   2. 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.
6. Determining the sign of the cofactor of an element of a matrix (chapter 6)
   1. 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.
   2. If the sum of the row and column the element is in is Even, the cofactor will be the Same as the minor.

Intermediate Value Theorem

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.