Polynomials
Contents
6. Polynomials#
Polynomials are the class of functions generated by adding, subtracting, and multiplying constants (real numbers) and (powers of) a variable \(x\).
6.1. Zoom In#
We’ve already been discussing some examples of polynomials. It’s typical to categorize polynomials by their degree.
Degree |
Polynomial |
Example |
|---|---|---|
0 |
\(c\) |
\(y = 7\) |
1 |
\(mx + b\) |
\(f(x) = 2x - 3\) |
2 |
\(ax^2 + bx + c\) |
\(y = x^2 - x + 1\) |
3 |
\(a_3 x^3 + a_2 x^2 + a_1 x + a_0\) |
\(g(x) = 3x^3 + x^2 - x\) |
\(\vdots\) |
\(\vdots\) |
|
\(n\) |
\(a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0\) |
In this table, all of the non-\(x\) letters (like \(c, m, a_0, a_1, ...\)) are taken to be some fixed constants. Note that those constants are allowed to be zero! If you look at \(g(x)\) in the table above, it has \(a_3 = 3\), \(a_2 = 1\), \(a_1 = -1\), and \(a_0 = 0\). You should pause to check that all of those make sense to you. These constants are called the coefficients of the polynomial. We normally write \(a_k\) to represent the coefficient on the term involving \(x^k\).
- polynomial#
A polynomial is any function that can be written in the form \(a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0\), where the numbers \(a_0, a_1, ..., a_n\) are constants and \(n\) is a non-negative integer.
- coefficient#
A coefficient of a term is its real number part (ignore the variable part). For example, the coefficient of \(\frac{5}{2}x^2\) is \(\frac{5}{2}\).
- degree#
The degree of a polynomial is the highest power of \(x\) that occurs in the polynomial (when the polynomial is written in general form).
Warning
You can only directly read the degree of a polynomial when it has been fully FOILed/expanded, with like terms collected. For example, if \(f(x) = 5(x-3)(x+4)\), then you might be tempted to say that the degree is 1. But if we FOIL, we get:
So the degree would be 2, and the leading coefficient would be 5.
Remark 6
The quadratics that we saw in the last section are the degree 2 polynomials. Something like \(y = 2x - 3\) is a degree 1 polynomial, because it can be thought of as \(y = 2x^1 - 3x^0\). Similarly, \(y = 7\) (a constant function) is a degree 0 polynomial, since it can be written \(7x^0\) (recall that \(x^0 = 1\)).
As in the previous section, we’re going to be focused on some ways that we can write a polynomial function.
General Form |
Factored Form |
|---|---|
\(a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0\) |
\(a_n (x - r_1)(x - r_2)\cdots (x - r_n)\) |
The left form is useful for determining what the “end behavior” of the graph looks like - where \(x\) is a large positive or negative number. The right form is more useful for determining the “inside” portion of the graph, because \(r_1, ..., r_n\) are the roots of the polynomial. Roots correspond to \(x\)-intercepts in the graph.
6.1.1. Determining End Behavior (The Leading Coefficient Test)#
First, we’re going to focus on determining what the edges of a polynomial’s graph should look like. We’ll learn how to determine the interior behavior of the graph later.
We saw in the last section that \(ax^2 + bx + c\) “opens up” if \(a\) is positive, and it “opens down” if \(a\) is negative. More generally, polynomials with an even degree will either “open up” or “open down”, depending on the positivity or negativity of their leading coefficient.
- leading coefficient#
The leading coefficient of a polynomial (in general form) is the coefficient of its highest-degree term.
For example, the coefficients of the terms in \(-x^4 + 2x^2 + 7\) are -1, 2, and 7. The leading coefficient would be -1, because that’s the coefficient on the term with the highest power of \(x\). This polynomial would open down on the edges of the graph by the Leading Coefficient Test (below).
6.1.1.1. The Leading Coefficient Test#
Coming Soon! See your lecture notes/videos for help with this topic.