Is a polynomial function a rational function?

Question

It's my first time on this forum so please forgive any errors in the presentation of my question.

My query refers to definitions of polynomials and rational functions. I have just started Ferrar's 'Higher Alegbra for Schools' and have become bogged down in the definitions. I have attached link but will try and explain myself first.

The first chapter begins with a number of definitions, in particular definitions of polynomial, rational and algebraic functions. Polynomials are defined strictly for positive exponents (n), whilst rational as the ratio of two polynomials. Mr Ferrar then goes on to describe and define algebraic functions. His example (see link) is a quadratic in $y$ with polynomials in $x$. He then goes on to define, in general, that algebraic functions are the same as the example but with rational functions of $x$.

Herein lies my lack of understanding. Put plainly my question is this,

Does a polynomial function imply rational?

By modern definitions it would appear so, but I am unable to see how Mr Ferrar's generality is to include polynomials as his definition seems inadequate.

Please forgive any ignorance on my part, I'm sure there's a subtlety here that I'm missing but I'm not quite sure what it is.

The book can be found here:

https://archive.org/details/higheralgebrafor0000ferr/page/4/mode/2up

Pages 2, 3 and 4 give details of my post.

Many thanks in advance for any advice given.

Answer 1

The link won't show pages 2 and 3 to non-subscribers (I'm guessing) so I can't tell what the book really says.

When you say polynomials are defined strictly for positive exponents, this is a bit vague. We would consider

$$3x^3 - 2x^2+\frac{1}{2}x -5$$

to be a polynomial, even though the last term is really $-5x^0$ and the exponent is not positive.

Therefore

$$1$$

is a polynomial and therefore

$$3x^3 - 2x^2+\frac{1}{2}x -5=\frac{3x^3 - 2x^2+\frac{1}{2}x -5}{1}$$

is a rational function.

I think this is the answer to your question: If a function is a polynomial, then it is also a rational function.

Answer 2

Consider the analogous case in geometry of the definition of points, line segments, and rectangles. A point has no length or area. A line segment has non zero length, but no area, and a rectangle has non zero length and width. These are regarded as three mutually exclusive concepts in classical geometry.

The situation in algebra is different. A constant is regarded as a polynomial of degree zero. A rational function is the quotient of two polynomials where either of them are allowed to be constant. Thus all polynomials are also considered to be rational functions and all constants are also considered to be polynomials.

Mr. Ferrar is probably working with a context closer to the geometric one described earlier. Thus constants, polynomials and rational functions are mutually exclusive. This is acceptable if the definitions are clear about the edge cases.