1. Roots (solutions) to a polynomial in a single variable
Quite why I was not taught this for ‘A’ level maths is beyond me (or more likely I was and have simply forgotten) but, if we have a polynomial in a single variable:
Then the general form of its factorisation is:
and it has the roots:
Here’s an (very) outline proof…
Take a polynomial with a known root then will divide evenly (no remainders) by and so we can say where is of one degree less than . We can continue this until we are left with a function of degree 0 – i.e. the constant (possibly 1) and then we have the form .
(And I recommend Elliptic Tales: Curves, Counting, and Number Theory for those who want to know more – I am on my second iteration of trying to read this, but it is good fun.)
2. I can construct a curve that has no tangent at any point
At least, algebraically, it has no tangent.
For instance, the line defined by .
We define the tangents for such lines, as, at the point as being of the form:
Now, , but note that and are also equal to zero for all points, so we have which describes no line.
But – geometrically – this line is the same as that described algebraically by which has a tangent line of .
Are they different lines? They are, algebraically.
(This comes from the same source as point 1).