Polynomials are a natural extension of quadratics that we just looked at. In its most general we have
$y = ax+b$ (using a for the slope)
$y = ax^2 + bx + c$
$y = a_n x^n + a_{n-1}x^{n-1} + ... + a_1 x + a_0$. Notice, it is made up of powers of x. It looks a little different because we don’t specify what $n$ is.
Now we want to understand the following about polynomials:
Polynomials are combinations of terms called monomials which look like $ax^n$ for some $a$ and $n$. With your graphing calculator, graph the following over the interval $[-2,2]$:
$$x,x^3,x^5,x^7$$
Then graph $x^2,x^4,x^6$
You should see that in the first case the graphs start from $-\infty$ and go to $+\infty$. What happens in the second case?
Next investigate the graphs of
$$x^3, -x^3, x^4, -x^4$$
The minus sign flips the graph.
Now, when you graph a polynomial in general, the leading term is the one with the highest power. ($a_n x^n$ in general). What you will learn is that if you zoom far enough out, that the graph of the polynomial will look like the graph of just the leading term. (Roughly) Try this out, by graphing both
$$x^3, x^3 - 3x$$
First on the interval $[-3,3]$ and then on the interval $[-30,30]$. Set your $y$-window to accomodate the whole graph (ymax = $30^3 = 27,000$.) See what I mean?
This observation is the basis of the leading coefficient test. It tells you the general shape of the graph.
. Of course, the other terms make a difference. Notice the difference between the graphs of $x^3$ and $x^3 - 3x$ when $x$ is in $[-3,3]$. The other terms give it wiggles. These wiggles can make the polynomial cross the $x$-axis more times. When a polynomial crosses the $x$-axis we call this a zero or root of the polynomial. In many mathematical problems, the answers are given in terms of roots of a polynomial. Knowing how to find these is important.
We should be able to find them with our calculator. (pretty much anyway). In section 2.3, there is a more systematic approach to this task.
This section introduces some theory into the solving of polynomials. But first, there are also a few skills to learn.
How can you divide $x-1$ into $x^2 + 3x + 2$? With long division! It works just like long division of numbers. Try it out. What it accomplishes is summarized in the following theorem that dates back to the greeks.
The remainder theorem. If we divide $p(x)$ by $q(x)$ we can uniquely find $r(x)$ of lower degree than $q(x)$ and $d(x)$ with
$$p(x) = q(x)d(x) + r(x)$$
To appreciate this, think about dividing numbers. We know that we can divide with remainder. For example 43 = 7*6 + 1. So dividing 43 by 7 leaves a remainder of 1.
If we divide by a term like $(x-c)$ (aka a linear factor) then the remainder must be of lower order so is just a number or a constant. In this case, there is a special way to do the division called synthetic division, that uses just the coefficients in a very compact manner. Try to figure it out from the examples in the book. It is very slick and gives you not only the remainder, but also $d(x)$ (the divisor)
Now back to factoring and zeroes. Here is the important fact.
A number $c$ is a root of $f(x)$ if $(x-c)$ is a factor of $f(x)$. That is $f(x) = (x-c)d(x)$ and the remainder is 0.
So factors and zeroes are similar, but not the same. (One is a number, the other a polynomial term.)