Binomial Expansion

Learn, then practise

The original worked lessons remain the main explanations. Use these focused reminders to check the conditions before you start.

Finite expansions, factorial notation and signs

Expand positive integer powers accurately.
Keep constant powers, signed factors and the requested order.
Distinguish a complete identity from a truncated expression.

For a positive integer \(n\), \((a+b)^n=\sum_{r=0}^{n}\binom nr a^{n-r}b^r\), where \(\displaystyle \binom nr=\frac{n!}{r!(n-r)!}\) and \(0!=1\). Apply the exponent to the whole signed term. "First terms" are not a complete expansion.

Read the full notes and 3 worked examples · Next: short practice

The general term, powers of x and missing terms

Match the requested exponent to an integer index.
Distinguish a coefficient, a term and its position.
Handle negative powers without substituting an excluded input.

For \((Ax^p+Bx^q)^n\), \(T_{r+1}=\binom nr A^{n-r}B^r x^{p(n-r)+qr}\). The index \(r\) must be an integer with \(0\le r\le n\). The fourth term uses \(r=3\). Negative powers require \(x\ne0\).

Read the full notes and 3 worked examples · Next: short practice

Coefficients in products and cancellation

Collect every pair of powers contributing to a target.
Simplify paired binomials before expanding where useful.
Use finite expansion and substitution to justify coefficient-sum results.

To obtain \(x^m\) in a product, add contributions whose exponents sum to \(m\). For example, the \(x^2\) coefficient can use \((0,2),(1,1),(2,0)\). Negative powers can require terms above degree \(m\) in the other factor. Any coefficient-sum or parity result must be derived, not recalled as a special theorem.

Read the full notes and 3 worked examples · Next: short practice

Unknown constants and an integer exponent

Translate coefficient conditions into simultaneous equations.
Retain both signs and any zero cases before division.
Check that an inferred exponent is an admissible integer.

For \((1+ax)^n\), the first coefficients are \(na\) and \(\displaystyle \frac{n(n-1)}2a^2\). Check nonzero conditions before dividing. A finite binomial exponent in this package is a positive integer. An even power usually leaves two possible signs.

Read the full notes and 3 worked examples · Next: short practice

Finite approximations, exact remainders and models

Choose a substitution and evaluate a stated truncation.
Calculate the signed remainder and absolute error.
Check a claimed accuracy and interpret units and percentage changes.

Write \(F(x)=P(x)+R(x)\), where \(P\) retains the requested terms. The signed remainder is \(R=F-P\); the absolute error is \(|R|\). If \(R>0\), \(P\) is low. A finite expansion has no convergence condition such as \(|x|<1\). Bounds and models here are algebraic applications; calculus is not needed.

Read the full notes and 3 worked examples · Next: short practice