General binomial expansion

Objective. Expand (1+x)^n for rational n within validity conditions.

1. Learn
2. Worked example
3. Practice
4. Feedback
5. Continue

Learn: the key idea

The general binomial expansion extends (1+x)ⁿ to rational or negative n using (1+x)ⁿ=1+nx+n(n−1)/2! x²+n(n−1)(n−2)/3! x³+..., valid only for |x|<1. Example: (1+x)^(1/2)≈1+(1/2)x−(1/8)x²+... for small |x|, useful for approximating square roots like √1.1 by setting x=0.1. Unlike the finite binomial theorem for positive integer n, this series never terminates, so a validity condition on x must always be stated.

Worked example

Expand (1+x)^(−2) using the general binomial expansion up to the term in x², stating the validity condition.

1 $Write n=-2 in (1+x)ⁿ=1+nx+n(n-1)/2! x²+...$
2 $Compute the x term: nx=-2x$
3 $Compute the x² term: n(n-1)/2! x²=(-2)(-3)/2 x²=3x²$
4 $Combine terms and state the validity condition |x|<1$

Answer: 1−2x+3x², valid for |x|<1

Concept mastery

Based on your past attempts at this lesson — the question types to practise again come first.

Complete a practice set to see your concept breakdown here.

Practice

Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.

Start practising

Common mistake. Students expand (1+x)ⁿ for negative or fractional n as if it were a finite polynomial, forgetting the series is infinite and only valid for |x|<1, which must be checked and stated.

Your progress

Tick each step as you finish it. Your exact reading position is saved automatically, so Continue drops you back on the same line.

0 of 5 steps complete

Your place saves automatically as you read.

My dashboard