Polynomials and Algebraic Functions in Edwards & Penney

presentation of Edwards & Penney's work, by Corry Shores
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Edwards & Penney's Calculus is an incredibly-impressive, comprehensive, and understandable book. I highly recommend it.

[the following will not stray from Edwards & Penney's procedure, so it is sometimes quotation.]

1.3 Polynomials and Algebraic Functions

Power Functions

Power functions take this form:

f (x) = x^k (where k is a constant)

In other words, power functions have exponents. Let's suppose that k = 0. Any number to the zeroth power is 1. Hence if k = 0, that yields the constant function

f (x) ≡ 1

The graph for a power function will take a certain shape. That shape depends on whether or not the exponent is even or odd.

Ex. 1

For example, when we graph functions with even-numbered exponents, they all "cup upward."

But if they are odd, they go "from southwest to northeast."

Combinations of Functions

We may combine functions to create new ones. Let's first suppose these things: f and g are functions, and c is a fixed real number. We now will find

cf : (scalar) multiple

f + g : sum

f - g : difference

f ∙ g : product

f / g : quotient

Ex:

Polynomials

A polynomial is a function that may be of a certain degree. If its degree is n, then we formulate it this way:

where the coefficients

a_0, a_1, . . . . . , a_n

are fixed real numbers, and also where

a_{n ≠ 0}

So we add a series of power functions in a polynomial. We obtain an nth-degree polynomial by summing the constant multiples of the power functions:

So we might have a polynomial to the first degree. It would then just be a linear function

a_1­­x + a₀

. Its graph would merely be a straight line. Second degree polynomials are quadratic functions.

Their graph is a parabola.

The zero solutions for function f (x) are the x-intercepts of the graph.

The fundamental theorem of algebra:

Every nth-degree polynomial has n zeros. Thus, an nth-degree polynomial has no more than n distinct real zeros.

Ex:

The diagrams before display polynomials.

These polynomials have the maximum number of real zeros that the fundamental theorem of algebra allows. Yet we saw with the high degree polynomial may only have a single real zero.

And in fact, the quadratic equation

has no zeros. A polynomial of an even degree can have any number of zeros, so long as they do not number more than its nth value. And one of an odd degree has any number of zeros between 1 and n. And outside its x-axis interval containing its zero values, the polynomial behaves "near infinity" just like the power functions (see the above diagrams).

from Edwards & Penney: Calculus. New Jersey: Prentice Hall, 2002, p.24-27.