Binomial expansion is used to expand expressions of the form \((a+b)^n\). In Year 13, this is especially important when the power is negative or fractional.
For any real number \(n\),
This expansion is valid for \(|x|<1\) when \(n\) is negative or fractional.
\[ (1+x)^n = 1 + nx + \frac{n(n-1)}{2}x^2 + \frac{n(n-1)(n-2)}{6}x^3 + \cdots \]
In many questions, you are asked only for the first 2, 3, or 4 terms.
If \(n\) is a positive integer, the expansion stops after a finite number of terms.
Example:
\[ (1+x)^3 = 1 + 3x + 3x^2 + x^3 \]
This is the usual binomial expansion from earlier algebra.
If \(n\) is negative, the expansion does not end. It becomes an infinite series.
Example: Expand \((1+x)^{-1}\)
\[ (1+x)^{-1} = 1 + (-1)x + \frac{(-1)(-2)}{2}x^2 + \frac{(-1)(-2)(-3)}{6}x^3 + \cdots \]
Notice that the signs alternate.
Expand \((1+2x)^{-2}\) up to the term in \(x^3\).
Using
\[ (1+u)^n = 1 + nu + \frac{n(n-1)}{2}u^2 + \frac{n(n-1)(n-2)}{6}u^3 + \cdots \]
with \(n=-2\) and \(u=2x\):
\[ (1+2x)^{-2} = 1 + (-2)(2x) + \frac{(-2)(-3)}{2}(2x)^2 + \frac{(-2)(-3)(-4)}{6}(2x)^3 + \cdots \]
\[ = 1 - 4x + 3(4x^2) - 4(8x^3) + \cdots \]
The same formula works for fractional powers.
Example: Expand \((1+x)^{1/2}\) up to the term in \(x^3\).
\[ (1+x)^{1/2} = 1 + \frac12 x + \frac{\frac12(-\frac12)}{2}x^2 + \frac{\frac12(-\frac12)(-\frac32)}{6}x^3 + \cdots \]
Often you must first rewrite the expression into the form \((1+u)^n\).
Example: Expand \((2-x)^{-1}\) up to the term in \(x^2\).
First factor out 2:
\[ (2-x)^{-1} = \left[2\left(1-\frac{x}{2}\right)\right]^{-1} \]
\[ = \frac12\left(1-\frac{x}{2}\right)^{-1} \]
Now use the expansion for \((1+u)^{-1}\):
\[ \left(1-\frac{x}{2}\right)^{-1} = 1 + \frac{x}{2} + \frac{x^2}{4} + \cdots \]
The general term in the expansion of \((1+x)^n\) is:
This is useful when finding a specific term.
Binomial expansions are often used for approximation when \(x\) is small.
Example:
\[ (1+x)^{-1} \approx 1-x \]
when \(|x|\) is very small.