Class 13 Notes

I. Polynomials
   A. Defintion
   B. Standard form
II. Solving and factoring polynomial equations
III. Graphs
     A. Limits
     B. Zeros


I. Polynomials

A. Definition

In this last week we will tackle a final kind of function, called a
polynomial.

A polynomial is a function f(x) that can be written as a sum of terms
that involve x raised to a non-negative integer power. For example

f(x) = x^2

is a polynomial, because it involves x raised to the power 2.

Similarly,

f(x) = 1 + x^2 + 2 x^3

is a polynomial, because all the terms being summed involve x raised
to a non-negative power. Notice that the term 1, with no x in it, is x
raised to the zero.

In contrast

f(x) = sqrt(2)
f(x) = x^-2
f(x) = e^x

are all non-polynomials.


B. Standard form

A polynomial can be written in many ways, but one of the most useful
is called standard form.

Standard form just means that the terms are ordered from the highest
power to the lowest power. Thus for example

f(x) = 1 + x^2 + x^3

is not in standard form, while

f(x) = x^3 + x^2 + 1

is the same polynomial in standard form. In general standard form
looks like

f(x) = a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + 
     ... + a_2 x^2 + a_1 x + a_0

where the a's are numbers, called coefficients.

The highest power of x present is called the degree of the polynomial,
so for example

f(x) = 2 x^4 + 2 x - 1

is a polynomial of degree 4.

Polynomials of degree 1 are sometimes called linear, those of degree 2
are called quadratic, those of degree 3 are cubic, degree 4 is
quartic, degree 5 is quintic, etc.

The entire term containing the highest power present is called the
leading term. In this example, the leading term is 2 x^4. Similarly, 2
is the leading coefficient.


II. Solving and factoring polynomial equations

Equations involving polynomials can sometimes be solved by
factoring. To take an example

x^4 - 5x^2 + 4 = 0

is an equation involving a polynomial.

We can factor this to 

(x^2 - 4)(x^2 - 1) = 0

Each of these factors is a difference of squares and can be factored further:

(x - 2)(x + 2)(x - 1)(x + 1) = 0

Clearly this can be solved by making any one of these four factors
zero, so there are four solutions: x = -2, +2, -1, +1.

The values of x for which a polynomial p(x) is equal to zero are
called its roots.

This illustrates a general principle: if you have an equation in the
form p(x) = 0, where p(x) is a polynomial, if you can factor the
polynomial than you can read off the answers.

Note that, in our example, the polynomial was of degree 4, and we
found 4 factors. This is another general principle: the number of
factors is equal to the degree of the polynomial.

It is easy to see why this should be, since every factor produces a
term with one more power of x.

The quadratic formula, we see, is a special case of this for
polynomials of degree 2. It gives us the factors.

It turns out that there are equivalent formulas for cubic polynomials
and even quartic ones, but there is no such formula for polynomials of
degree 5 or higher. It is possible to prove this rigorously.

Thus we cannot in general factor high degree polynomials to find exact
solutions. Instead, we are reduced to finding approximate solutions
using computers.

Note that factors are sometimes repeated. Thus for example the
polynomial

p(x) = x^2 - 4x + 4

factors to 

p(x) = (x-2)(x-2).

This polynomial has two roots that are not distinct, they are both
2. This is called a root of multiplicity 2.

We can turn this factoring around: given a set of roots, we can always
construct the polynomial that has those roots, just by finding the
corresponding factors.

Thus if we seek a polynomial with roots 1, 2, and 3, we have

p(x) = (x-1)(x-2)(x-3)
     = (x^2 - 3x + 2)(x - 3)
     = x^3 - 6x^2 - 7x - 6


III. Graphs

A. Limiting behavior

We can sketch out what graphs of polynomials look like by considering
two features of their behavior: their roots and their leading terms.

Let's take the example we just used:

p(x) =  x^3 - 6x^2 - 7x - 6

When x is very large, the x^3 term will be much larger than any of the
other ones, because x^3 is always bigger than x^2 if we pick x
sufficiently large.

Thus at large values of x, the overall shape of the function will be
determined by the leading term. The same goes for at x << 0, i.e. for
very negative values of x.

Thus the long-run behavior of polynomial functions only depends on
their leading terms.

In the example we have, x^3 is positive when x is positive, and
negative when x is negative, so we expect the graph to be positive at
large x and negative at small x.

If the sign of the x^3 term were reversed it would be the other wa
around: negative when x was large, positive when it was small.

If we instead had

p(x) ~ x^4,

however, we notice that x^4 is always positive, whether x is positive
or negative. Thus at large or small x the polynomial is always
positive.

If the sign were opposite, say p(x) ~ -2 x^4, then at large or small x
the polynomial would always be negative.

To sum up, for a polynomial p(x) whose leading term is a_n x^n, we have:

- If n is even and a_n > 0, p(x) is positive and increasing toward
  infinity for both x >> 1 and x << 1

- If n is even and a_n < 0,  p(x) is negative and decreasing toward
  negative infinity for both x >> 1 and x << 1

- If n is odd and a_n > 0, p(x) is positive and increasing toward
  infinity for both x >> 1, and negative and decreasing toward
  negative infinity for x << 1

- If n is odd and a_n < 0, p(x) is positive and increasing toward
  infinity for both x << 1, and negative and decreasing toward
  negative infinity for x >> 1


B. Zeros

We also know what the graph looks like in the vicinity of roots of a
polynomial.

Suppose we have a root at x = k, so the function looks like

p(x) = (x - k)^n . other factors,

where n indicates the multiplicity of the root.

Clearly p(x) = 0 at x = k, so the graph goes to zero at the
root. 

The shape near the root depends on n. To see how it behaves, consider
what happens if we pick x to be a little bigger than k or a little
smaller than k.

Suppose first that n is odd, for example n = 1. If x > k, then (x-k)^n
is positive, and the sign of p(x) depends on whether the other factors
are positive or negative. Conversely, for x < k, then (x-k)^n is
negative, and the sign of p(x) is the opposite of whatever sign the
other factors come out to have. This will be the same sign as at x >
k, provided we choose x just barely to the left or right of k.

In this case, the sign of p(x) is opposie for x slightly less than k
than for x slightly greater than k, so p(x) passes through zero at
x=k, and is positive on one side and negative on the other.

Now suppose that n is even, so the root is of even multiplicity. In
this case (x-k)^n is positive no matter whether we choose x just
bigger or just smaller than k.

As a result, we see that p(x) must hit the axis, but not pass through
it. Instead, it "bounces off" the axis.

Given this information, we can sketch a graph of a polynomial just
given its zeros and leading coefficient.

Example: sketch graph of (x - 3)(x + 1)^2(x-1)

Practice problem: sketch graphs of 

(x - 4)^3 (x - 3)^4 (x - 2)^5