SignificancePolynomial functions are relatively easy to understand. Low degree polynomial equations can be solved explicitly. Polynomials provide good examples for studying more general functions.
Standard FormsThe standard form of a polynomial is
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 BehaviorThe behavior of a polynomial function
f(x) = anxn + an-1xn-1 + . . . a1x + a0
is essentially the same as that of the monomial
over a large enough scale. To illustrate, this
consider the two functions
Exercise: Graph several polynomial functions together with their leading terms.
Here are a few examples:
Zoom out on these graphs and watch the graphs of the polynomials and the graphs of the leading terms coalesce.
Graphs of MonomialsThe general shape of the graph of a monomial
depends only on
As n gets larger, all graphs become
increasingly flat on the interval
Roots or Zeroes
A degree 2 polynomial is called a quadratic polynomial and can be written in the form
Its graph is a parabola. The quadratic formula states that
the roots of
Roots of a polynomial can also be found if you can factor the polynomial.