Binomial Theorem for Rational n

The Binomial Theorem for (1 + x)ⁿ

The previous version of the binomial theorem only works when n is a positive integer. If n is any fraction, the binomial theorem becomes:

PROVIDING ½x½ < 1

Note that while the previous series stops, this one goes on forever.

Example

Find the expansion of (5x + 2)^1/2

We need to transform this so it looks like (1 + x)^1/2, so lets take out a factor of 2:

(5x + 2)^1/2 = (2[5x/2 + 1])^1/2

Now, where we have ‘x’ in the above formula, we need 5x/2 and where we have n, we need ½ .

= Ö2(1 + 5x/2)^1/2
= Ö2[ 1 + ½ (5x/2) + ½ × ½ (- ½ )(25x²/4) + … ]

Remember, this is only valid if –1 < 5x/2 < 1, in other words, -2/5 < x < 2/5

Using Partial Fractions

We can expand more complicated expressions, now, using the method of partial fractions where appropriate.

Example

We can split this up, using partial fractions, into:

  1      +     1            .
1 + x       5x + 2

Now expand (1 + x)^-1 and (5x + 2)^-1 as described above and add.