# 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) = anxn + an-1xn-1 + . . . + a1x + a0

The ai are real numbers and are called coefficients. The term an 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) = anxn + an-1xn-1 + . . . a1x + a0

is essentially the same as that of the monomial

anxn

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) = anxn

depends only on

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

2. The sign of an.

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.

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