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O-Level A-Math and SEC G3 Additional Mathematics K341

A5: Binomial expansions

Use the positive-integer binomial theorem and general term to identify a required term, power, or coefficient.

Download this chapter PDF Download complete Additional Mathematics PDF

Core notes

The binomial theorem expands $(a+b)^n$ for a positive integer $n$ without repeated multiplication. Factorial notation defines the coefficients, while the general term locates a requested power or coefficient directly.

Binomial theorem for positive integer powers

For a positive integer $n$ , the expansion of $(a+b)^n$ has $n+1$ terms. The power of $a$ decreases from $n$ to $0$ , while the power of $b$ increases from $0$ to $n$ .

Factorial and combination notation

Factorial notation is $n!=n(n-1)\cdots2\cdot1$ , with $0!=1$ . The combination $\binom nr$ counts selections of $r$ objects from $n$ and supplies the coefficient of the term indexed by $r$ .

The general term

The term indexed by $r$ is $\binom nr a^{n-r}b^r$ , where $0\leq r\leq n$ . If either binomial term contains $x$ , combine all powers of $x$ , solve for the required exponent, and then evaluate the coefficient.

Formulae and methods

Relationship	Use
$(a+b)^n=\sum_{r=0}^{n}\binom nr a^{n-r}b^r$	Expand a positive integer power.
$\binom nr=\frac{n!}{r!(n-r)!}$	Calculate a binomial coefficient.
$T_{r+1}=\binom nr a^{n-r}b^r$	Find the term indexed by $r$ .

Worked examples

Example 1: Find the coefficient of $x^3$ in $(2+x)^5$ .

Use the general term $\binom5r2^{5-r}x^r$ .
For $x^3$ , set $r=3$ .
Calculate $\binom532^2=10\times4$ .

Answer: The coefficient is $40$ .

Example 2: Find the term independent of $x$ in $(x^2+\frac2x)^6$ .

The general term is $\binom6r(x^2)^{6-r}(2x^{-1})^r$ .
The power of $x$ is $12-3r$ ; set $12-3r=0$ , so $r=4$ .
Evaluate $\binom642^4=15\times16$ .

Answer: The independent term is $240$ .

Chapter checkpoint

Confirm that the exponent is a positive integer before using this syllabus form of the theorem.
Index terms consistently so the first term corresponds to $r=0$ .
Combine the coefficient, power of the first term, and power of the second term before extracting a requested power.

Exam traps and retrieval check

Avoid these traps

Calling the term with index $r$ the $r$ th term instead of the $(r+1)$ th term.
Changing the coefficient but not the associated powers.
Assuming the syllabus requires greatest-term or coefficient-property results that it explicitly excludes.

Check from memory

How many terms are in

(a+b)^n

$n+1$ terms.

What is

0!

$1$ .

What range can

r

take in the general term?

The integers from $0$ to $n$ , inclusive.

Official K341 coverage

This chapter maps to 3 official outcomes: A5.1, A5.2, A5.3. Check the official 2027 K341 syllabus for exact assessable wording and paper details.

Loading blog content…

O-Level A-Math and SEC G3 Additional Mathematics K341

A5: Binomial expansions

Use the positive-integer binomial theorem and general term to identify a required term, power, or coefficient.

Download this chapter PDF Download complete Additional Mathematics PDF

Core notes

Binomial theorem for positive integer powers

For a positive integer $n$ , the expansion of $(a+b)^n$ has $n+1$ terms. The power of $a$ decreases from $n$ to $0$ , while the power of $b$ increases from $0$ to $n$ .

Factorial and combination notation

Factorial notation is $n!=n(n-1)\cdots2\cdot1$ , with $0!=1$ . The combination $\binom nr$ counts selections of $r$ objects from $n$ and supplies the coefficient of the term indexed by $r$ .

The general term

Formulae and methods

Relationship	Use
$(a+b)^n=\sum_{r=0}^{n}\binom nr a^{n-r}b^r$	Expand a positive integer power.
$\binom nr=\frac{n!}{r!(n-r)!}$	Calculate a binomial coefficient.
$T_{r+1}=\binom nr a^{n-r}b^r$	Find the term indexed by $r$ .

Worked examples

Example 1: Find the coefficient of $x^3$ in $(2+x)^5$ .

Use the general term $\binom5r2^{5-r}x^r$ .
For $x^3$ , set $r=3$ .
Calculate $\binom532^2=10\times4$ .

Answer: The coefficient is $40$ .

Example 2: Find the term independent of $x$ in $(x^2+\frac2x)^6$ .

The general term is $\binom6r(x^2)^{6-r}(2x^{-1})^r$ .
The power of $x$ is $12-3r$ ; set $12-3r=0$ , so $r=4$ .
Evaluate $\binom642^4=15\times16$ .

Answer: The independent term is $240$ .

Chapter checkpoint

Confirm that the exponent is a positive integer before using this syllabus form of the theorem.
Index terms consistently so the first term corresponds to $r=0$ .
Combine the coefficient, power of the first term, and power of the second term before extracting a requested power.

Exam traps and retrieval check

Avoid these traps

Calling the term with index $r$ the $r$ th term instead of the $(r+1)$ th term.
Changing the coefficient but not the associated powers.
Assuming the syllabus requires greatest-term or coefficient-property results that it explicitly excludes.

Check from memory

How many terms are in

(a+b)^n

$n+1$ terms.

What is

0!

$1$ .

What range can

r

take in the general term?

The integers from $0$ to $n$ , inclusive.

Official K341 coverage

This chapter maps to 3 official outcomes: A5.1, A5.2, A5.3. Check the official 2027 K341 syllabus for exact assessable wording and paper details.

Core notes

Binomial theorem for positive integer powers

Factorial and combination notation

The general term

Formulae and methods

Worked examples

Example 1: Find the coefficient of x3x^3x3 in (2+x)5(2+x)^5(2+x)5.

Example 2: Find the term independent of xxx in (x2+2x)6(x^2+\frac2x)^6(x2+x2​)6.

Chapter checkpoint

Exam traps and retrieval check

Avoid these traps

Check from memory

Official K341 coverage

Core notes

Binomial theorem for positive integer powers

Factorial and combination notation

The general term

Formulae and methods

Worked examples

Example 1: Find the coefficient of x3x^3x3 in (2+x)5(2+x)^5(2+x)5.

Example 2: Find the term independent of xxx in (x2+2x)6(x^2+\frac2x)^6(x2+x2​)6.

Chapter checkpoint

Exam traps and retrieval check

Avoid these traps

Check from memory

Official K341 coverage

Example 1: Find the coefficient of $x^3$ in $(2+x)^5$ .

Example 2: Find the term independent of $x$ in $(x^2+\frac2x)^6$ .

Example 1: Find the coefficient of $x^3$ in $(2+x)^5$ .

Example 2: Find the term independent of $x$ in $(x^2+\frac2x)^6$ .