Polynomial Functions

Significance

Polynomial functions are relatively easy to understand. Low degree polynomial equations can be solved explicitly. Polynomials provide good examples for studying more general functions.

Standard Forms

The standard form of a polynomial is

f(x) = a_nxⁿ + a_n-1x^n-1 + ^{. . .} + a₁x + a₀

The a_i are real numbers and are called coefficients. The term a_n is assumed to be non-zero and is called the leading term. The degree of the polynomial is the largest exponent of x which appears in the polynomial -- it is also the subscript on the leading term.

A polynomial with one term is called a monomial.

A degree 0 polynomial is a constant. A degree 1 polynomial is a linear function, a degree 2 polynomial is a quadratic function, a degree 3 polynomial a cubic, a degree 4 a quartic, and so on.

Large Scale Behavior

The behavior of a polynomial function

f(x) = a_nxⁿ + a_n-1x^n-1 + ^{. . .} a₁x + a₀

is essentially the same as that of the monomial

a_nxⁿ

over a large enough scale. To illustrate, this consider the two functions f(x) = 3 x³ - 5 x² + x +1 and g(x) = 3 x³. By clicking on the following links you can see these graphed on the indicated scales. See what happens as you go from the smallest to the largest scales:

-1 < x < 1

-10 < x < 10

-100 < x < 1000

Exercise: Graph several polynomial functions together with their leading terms.

Here are a few examples:

y = -(1/2)x² + x + 3,
y = 2 x³ + x² - 1,
y = - x⁶ - x⁴ + x³ - 1.

Zoom out on these graphs and watch the graphs of the polynomials and the graphs of the leading terms coalesce.

Graphs of Monomials

The general shape of the graph of a monomial

f(x) = a_nxⁿ

depends only on

The parity of n, i.e., whether n is even or odd.
The sign of a_n.

As n gets larger, all graphs become increasingly flat on the interval |x| < 1 and increasingly steep for |x| > 1. All polynomial functions grow without bound in a positive or negative direction for large and large negative x.

Characterisitcs of Graphs of a_n xⁿ
	n EVEN See Graphs	n ODD See Graphs
a_n > 0	Always positive. Resembles parabola opening up.	Negative for x < 0 Positive for x > 0 Resembles graph of x³.
a_n < 0	Always negative. Resembles parabola opening down.	Positive for x < 0 Negative for x > 0 Resembles graph of -x³.

Roots or Zeroes

If a polynomial f(x) has a root r, then f(x) can be factored as

f(x) = (x - r)g(x)

for some polynomial g(x). This precludes an n^th degree polynomial from having more than n roots. For example, if a quadratic polynomial had three roots, we would have
a x² + b x + c = (x - r₁)(x - r₂)(x - r₃)g(x).
But this is not possible since the right-hand side contains at least an x³ term.

A root (or zero) of a polynomial f(x) is a number r such that f(r)=0. A polynomial of degree n can have at most n distinct roots.

A degree 2 polynomial is called a quadratic polynomial and can be written in the form

f(x) = a x² + b x + c.

Its graph is a parabola. The quadratic formula states that the roots of a x² + b x + c = 0 are given by

Roots of a polynomial can also be found if you can factor the polynomial.

Examples and calculations are on the next page.

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