Rational-index binomial expansions, partial fractions and polynomial division

Last time

Before we start: last time, in The remainder and factor theorems and cubic equations, you learned these points. Here they are again on one card. Keep them in mind, because today builds on them.

Expanding what cannot be multiplied out

How would you write the square root of one plus four x as a polynomial? You cannot multiply it out, because it has no whole number power. But you can write it as an infinite series in ascending powers of x, and a fraction can be split into simpler pieces first. Those are the two tricks in this lesson.

The expansion for any index

For any rational index n, one plus x to the n equals one, plus n x, plus n times n minus one over two factorial x squared, and so on. When n is a positive whole number, the series stops. For any other n, it goes on for ever. It only works when x is small enough: the modulus of x must be less than one. This is the validity condition.

Expressions with a number in front

To expand something like two plus x to the power minus one, write it as two times, one plus x over two, to the minus one. The expansion then applies to x over two. The condition becomes the modulus of x over two less than one: the modulus of x less than two. Do not forget to multiply the whole series by the power of the constant you took out.