What is the general term of a binomial expansion?

T(r + 1) = C(n, r) aⁿ⁻ʳ bʳ, for r = 0, 1, …, n. It lets you pick out one term, such as the constant term or the x³ term, without writing the whole expansion.

Does DSE M2 include the binomial theorem for negative or fractional powers?

No. The M2 syllabus covers positive integral indices only. Expansions of trinomials, the greatest coefficient or greatest term, and numerical approximation are also listed as not required.

How is Pascal’s triangle related to the binomial theorem?

Row n of the triangle lists the coefficients C(n, 0), C(n, 1), …, C(n, n) of (a + b)ⁿ. Each entry is the sum of the two above it. In Chinese it is called Yang Hui’s triangle.

The Binomial Theorem (DSE M2): Expansion and General Term

For a positive integer n, (a + b)ⁿ = Σ C(n, r) aⁿ⁻ʳ bʳ for r = 0 to n. The general term is T_r+1 = C(n, r) aⁿ⁻ʳ bʳ. To find a particular coefficient, write the general term, collect the power of x, and solve for r.

Solve The Binomial Theorem questions with AI

Pascal’s (Yang Hui’s) triangle up to n = 6. Row 5 gives the coefficients 1, 5, 10, 10, 5, 1 used in Example 1. Try it yourself →

Binomial Theorem Formulas

Expansion(a + b)ⁿ = C(n, 0)aⁿ + C(n, 1)aⁿ⁻¹b + … + C(n, n)bⁿ

General termT_r+1 = C(n, r) aⁿ⁻ʳ bʳ

Binomial coefficientC(n, r) = n! / (r!(n − r)!)

Special case(1 + x)ⁿ = 1 + nx + [n(n − 1)/2]x² + … + xⁿ

Sigma notationΣ(ax_r + by_r) = aΣx_r + bΣy_r, Σ_i=1ⁿ a = na

Worked Examples — The Binomial Theorem

Example 1: Finding a coefficient

Find the coefficient of x³ in the expansion of (2 − x)⁵.

T_r+1 = C(5, r) 2⁵⁻ʳ (−x)ʳ = C(5, r) 2⁵⁻ʳ (−1)ʳ xʳ
For x³, r = 3
C(5, 3) × 2² × (−1)³ = 10 × 4 × (−1) = −40

Answer: −40

Example 2: Finding the constant term

Find the constant term in the expansion of (x² + 1/x)⁶.

T_r+1 = C(6, r) (x²)⁶⁻ʳ (x⁻¹)ʳ = C(6, r) x¹²⁻³ʳ
Constant term: 12 − 3r = 0, so r = 4
C(6, 4) = 15

Answer: 15

💡 Exam Traps & Secrets

⚠️ The sign belongs to the termIn (2 − x)⁵ the second term is −x, so the general term contains (−1)ʳ. Dropping it makes every odd-power coefficient the wrong sign.

⚠️ Term number is r + 1The term with C(n, r) is the (r + 1)th term, because r starts from 0. The "4th term" uses r = 3.

✏️ Self-Test: Quick Interactive Practice

How to study: Click any question below to instantly load it on the coordinate grid and check your steps.

(Expansion)Expand (1 + 2x)⁴ in ascending powers of x.

(Product of expansions)Find the coefficient of x² in the expansion of (1 + x)⁴(1 − 2x)³.

(Constant term)Find the constant term in the expansion of (2x − 1/x²)⁹.