Definition
A polynomial (German: ganzrationale Funktion) in x is a function that is made up of a finite number of terms of the form $a_nx^n$ where $a_n$ is a constant and n is a nonnegative integer.
It can be written in two forms:
$$f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$$
or
$$f(x) = a_0 + a_1x + ... + a_{n-1}x^{n-1} + a_nx^n $$
depending in what order you want the powers of x to be.
examples
roots
there are methods to find the roots for 2nd, 3rd and 4th degree.
For higher degrees, the French mathematician Evariste Galois (1811-1832) proved that it is not posssible to express the solutions of a fith-degree or higher in terms of its coefficients using algebraic operations. Footnote (Howard Anton, Calculus)
factorising
the null factor law
generally
$${A \cdot B = 0}$$ implies either A = 0 or B = 0
We use this fact to solve a variety of problems.
- quadratics in factored form
- polynomials
- exponential functions and their derivatives, extrema and points of inflexion
tips and tricks in factorising
cubic equation
$ax^3 + bx^2 + cx + d = 0$
$(x- \alpha)(x - \beta)(x - \gamma) = 0$
remainder theorem
If P(x) is divided by (x-c) then P(c) is the remainder.
and $P(x) = (x - c) Q(x) + R(c)$
examples
factor theorem
The binomial (x-c) is a factor of P(x) if and only if $P(c) = 0$.
examples
polynomial division
If we know one of the roots of a polynomial, say $x_1$, we can carry out long division to find the remaining polynomial Q(x).
$${P(x) = Q(x) \cdot (x-x_1)}$$
the method
$${\begin{align} & x^3 & - 6x^2 & +11x & - 6 & : & x - \color{violet}{1} = x^2 \color{orange}{- 5x} + \color{maroon} {6} \newline & x^3 & - x^2 & & &\newline & & -5x^2 & + 11x \newline & & \color{orange}{- 5x^2} & +\color{orange}{ 5x} & & \newline & & & + 6x & -6 \newline & & & + \color{maroon}{ 6x} & - \color{maroon}{6} \newline & & & & 0 \end{align}}$$
synthetic division (Ruffini’s rule)
(x - 1) leads to +1 on the left hand side as x = 1 is the root.
$${\begin{align} & +1 & {-6} && +11 && -6 & \newline \color{violet}{+1} ~ | & ~ ~ 0 & 1 && -5 && 6 \newline & ~ ~ 1 & \color{orange}{-5} && + \color{maroon}{6} && 0 \end{align}}$$
see also Wolfram MathWorld
construction method cubic roots
TODO - article from Mathelehren