How Do You Extend the Binomial Theorem to Fractional or Negative Exponents?

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This is AHL 1.10 content in IB Mathematics AA HL and can be examined on Paper 1 or Paper 2.

The generalized binomial coefficient defines every term:

Substituting this coefficient gives the reusable form:

Here, $r$ is a non-negative integer. If $n$ is not a non-negative integer, no factor necessarily becomes zero, so the expansion generally continues indefinitely rather than terminating as a polynomial.

For a binomial not initially written as $1+x$, factor out the first term before expanding:

The convergence condition must be applied to the new variable. Thus, require $\left|\frac{b}{a}\right|<1$, not merely $|b|<1$. This step is essential because the generalized series is local, not an identity valid for every real $x$. Always check that the chosen substitution satisfies this restriction.

Example: approximate $\sqrt{1.04}$ using terms through $x^2$. Choose $n=\frac{1}{2}$ and $x=0.04$:

The exact truncated calculation gives $1.0198$, which is close to the calculator value. The omitted terms create the approximation error; including more terms normally improves accuracy because $|0.04|<1$.

The common misconception is treating a fractional or negative expansion as a finite polynomial. It is an infinite series, and its convergence condition must accompany the expansion. Also, do not use ordinary integer-only combinations on a calculator; calculate coefficients from the generalized product.

In an IB response, first rewrite the expression in the form $A(1+u)^n$, then state $|u|<1$, write enough terms or the general term, substitute $n$, and simplify each coefficient. For a numerical approximation, show the substituted series before giving the decimal answer. Paper 1 usually rewards exact algebra without a calculator, whereas Paper 2 may use the series for approximation. A frequent mark loss is omitting the interval of convergence or stopping without an ellipsis, which incorrectly suggests a finite result.