Polynomial functions

Abstract

We now proceed to study polynomial functions, which are fundamental in Calculus and Linear Algebra. A polynomial function or polynomial, f (x) has the form

$$f\left( x \right) = {a_0} + {a_1}x + {a_2}{x^2} + {a_3}{x^3} + \cdots + {a_n}{x^n},$$

(10.1)

where a _o , a_l, ... , a_n, are given rational numbers called the coefficients and the variable x varies over some set of rational numbers. The value of a polynomial function f(x) can be directly computed by adding and multiplying rational numbers. The Dinner Soup function f (x) = 15x is an example of a linear polynomial with n = 1, ao = 0 and a_i = 15, and the Muddy Yard function f (x) = x ² is an example of a quadratic function with n = 2, a_o = a_l = 0, and a₂ = 1.

Sometimes he thought to himself, “Why?” and sometimes he thought, “Wherefore?”, and sometimes he thought, “Inasmuch as which?”. (Winnie-the Pooh)

He was one of the most original and independent of men and never did anything or expressed himself like anybody else. The result was that it was very difficult to take notes at his lectures so that we had to trust mainly to Rankine’s text books. Occasionally in the higher classes he would forget all about having to lecture and, after waiting for ten minutes or so, we sent the janitor to tell him that the class was waiting. He would come rushing into the door, taking a volume of Rankine from the table, open it apparently at random, see some formula or other and say it was wrong. He then went up to the blackboard to prove this. He wrote on the board with his back to us, talking to himself, and every now and then rubbed it all out and said it was wrong. He would then start afresh on a new line, and so on. Generally, towards the end of the lecture he would finish one which he did not rub out and say that this proved Rankine was right after all. (Rayleigh about Reynolds)