IB Math AA SL Topic 1.9: Binomial theorem | Notes & Flashcards | Aimnova

IB Math AA SL — Binomial theorem

Topic 1.9 of IB Mathematics: Analysis and Approaches covers Binomial theorem, which is part of Unit 1: Number & Algebra. Students explore key concepts including Pascal's triangle & nCr, Binomial expansion, Finding a term. A strong understanding of binomial theorem is essential for IB Math AA SL exams and builds the foundation for connected topics across the syllabus.

Key concepts in Binomial theorem

Key Idea: The binomial theorem expands a power of a two-term bracket, (a + b)ⁿ, without multiplying it all out — and lets you grab one term straight away. It shows up on both papers, often as find the coefficient of xᵏ or find the constant term.

🔢 The formulas you're given

\binom{n}{r} = \frac{n!}{r!\,(n - r)!}

$n$: the power of the bracket (top of ⁿCᵣ)
$r$: which term — 0 for the first, counting up

(a + b)^n = a^n + \binom{n}{1}a^{n-1}b + \cdots + \binom{n}{r}a^{n-r}b^r + \cdots + b^n

$a^{n-r}$: first term — its power **falls** n → 0
$b^{r}$: second term — its power **rises** 0 → n

(a + b)ⁿ has n + 1 terms, and in each term the two powers sum to n. Need just one term? Use the general term ⁿCᵣ aⁿ⁻ʳ bʳ: set the exponent of x equal to the power you want, solve for r, then compute that single coefficient — no full expansion.

✏️ IB-style worked examples

IB-style question — expand a bracket (Paper 1)

Expand (x + 2)⁴ without a calculator.

Step by step:

Row 4 of Pascal's triangle gives 1, 4, 6, 4, 1; powers of x fall, powers of 2 rise.
$x^4 + \binom{4}{1}x^3(2) + \binom{4}{2}x^2(2^2) + \binom{4}{3}x(2^3) + 2^4$
Work out each coefficient (2² = 4, 2³ = 8, 2⁴ = 16).
$= x^4 + 8x^3 + 24x^2 + 32x + 16$

Final answer:

(x + 2)⁴ = x⁴ + 8x³ + 24x² + 32x + 16 (5 terms; powers sum to 4)

IB-style question — a coefficient with the nCr (Paper 2)

Find the coefficient of x³ in the expansion of (2x − 1)⁵.

Step by step:

General term: ⁵Cᵣ (2x)⁵⁻ʳ (−1)ʳ. The power of x is 5 − r, so for x³ take r = 2.
$\binom{5}{2}(2x)^{3}(-1)^{2}$
Use the GDC for ⁵C₂ = 10, then cube the whole 2x and square the −1.
$= 10 \times 8x^3 \times 1 = 80x^3$

Final answer:

Coefficient of x³ = 80

IB-style question — find an unknown constant

In the expansion of (x + k)⁶, the coefficient of x⁴ is 60. Find the possible values of k.

Step by step:

The x⁴ term: power of x is 6 − r = 4, so r = 2.
$\binom{6}{2}x^4 k^2 = 15k^2 x^4$
Set the coefficient equal to 60.
$15k^2 = 60 \;\Rightarrow\; k^2 = 4$
Solve — an even power gives both signs.
$k = \pm 2$

Final answer:

k = ±2

Important: A coefficient or minus sign is part of the term, so raise the whole thing: (2x)³ = 8x³ (not 2x³), (−3)² = +9 (not −9). And "term independent of x" / "constant term" means the power of x is 0 — set the exponent to 0 and solve for r.

Tap each card to reveal the answer.

Exam Tips

Coefficients of (a + b)ⁿ = row n of Pascal's triangle = ⁿC₀, ⁿC₁, …, ⁿCₙ.
Paper 1: small powers by hand. Paper 2: ⁿCᵣ via MATH → ▶ PRB → 3:nCr.
General term ⁿCᵣ aⁿ⁻ʳ bʳ — match the exponent to the power you want to get r.
Bracket the whole term before raising: (2x)³ = 8x³, (−3)² = +9; constant term ⇒ power of x is 0.
Coefficient given → set it equal and solve (even power ⇒ ±); n unknown → use ⁿC₂, a quadratic in n.

IB Math AA SL — Binomial theorem

Key concepts in Binomial theorem

Key Idea: The binomial theorem expands a power of a two-term bracket, (a + b)ⁿ, without multiplying it all out — and lets you grab one term straight away. It shows up on both papers, often as find the coefficient of xᵏ or find the constant term.

🔢 The formulas you're given

\binom{n}{r} = \frac{n!}{r!\,(n - r)!}

$n$: the power of the bracket (top of ⁿCᵣ)
$r$: which term — 0 for the first, counting up

(a + b)^n = a^n + \binom{n}{1}a^{n-1}b + \cdots + \binom{n}{r}a^{n-r}b^r + \cdots + b^n

$a^{n-r}$: first term — its power **falls** n → 0
$b^{r}$: second term — its power **rises** 0 → n

(a + b)ⁿ has n + 1 terms, and in each term the two powers sum to n. Need just one term? Use the general term ⁿCᵣ aⁿ⁻ʳ bʳ: set the exponent of x equal to the power you want, solve for r, then compute that single coefficient — no full expansion.

✏️ IB-style worked examples

IB-style question — expand a bracket (Paper 1)

Expand (x + 2)⁴ without a calculator.

Step by step:

Row 4 of Pascal's triangle gives 1, 4, 6, 4, 1; powers of x fall, powers of 2 rise.
$x^4 + \binom{4}{1}x^3(2) + \binom{4}{2}x^2(2^2) + \binom{4}{3}x(2^3) + 2^4$
Work out each coefficient (2² = 4, 2³ = 8, 2⁴ = 16).
$= x^4 + 8x^3 + 24x^2 + 32x + 16$

Final answer:

(x + 2)⁴ = x⁴ + 8x³ + 24x² + 32x + 16 (5 terms; powers sum to 4)

IB-style question — a coefficient with the nCr (Paper 2)

Find the coefficient of x³ in the expansion of (2x − 1)⁵.

Step by step:

General term: ⁵Cᵣ (2x)⁵⁻ʳ (−1)ʳ. The power of x is 5 − r, so for x³ take r = 2.
$\binom{5}{2}(2x)^{3}(-1)^{2}$
Use the GDC for ⁵C₂ = 10, then cube the whole 2x and square the −1.
$= 10 \times 8x^3 \times 1 = 80x^3$

Final answer:

Coefficient of x³ = 80

IB-style question — find an unknown constant

In the expansion of (x + k)⁶, the coefficient of x⁴ is 60. Find the possible values of k.

Step by step:

The x⁴ term: power of x is 6 − r = 4, so r = 2.
$\binom{6}{2}x^4 k^2 = 15k^2 x^4$
Set the coefficient equal to 60.
$15k^2 = 60 \;\Rightarrow\; k^2 = 4$
Solve — an even power gives both signs.
$k = \pm 2$

Final answer:

k = ±2

Important: A coefficient or minus sign is part of the term, so raise the whole thing: (2x)³ = 8x³ (not 2x³), (−3)² = +9 (not −9). And "term independent of x" / "constant term" means the power of x is 0 — set the exponent to 0 and solve for r.

Tap each card to reveal the answer.

Exam Tips

Coefficients of (a + b)ⁿ = row n of Pascal's triangle = ⁿC₀, ⁿC₁, …, ⁿCₙ.
Paper 1: small powers by hand. Paper 2: ⁿCᵣ via MATH → ▶ PRB → 3:nCr.
General term ⁿCᵣ aⁿ⁻ʳ bʳ — match the exponent to the power you want to get r.
Bracket the whole term before raising: (2x)³ = 8x³, (−3)² = +9; constant term ⇒ power of x is 0.
Coefficient given → set it equal and solve (even power ⇒ ±); n unknown → use ⁿC₂, a quadratic in n.

IB Math AA SL — Binomial theorem

Key concepts in Binomial theorem

🔢 The formulas you're given

✏️ IB-style worked examples

IB-style question — expand a bracket (Paper 1)

IB-style question — a coefficient with the nCr (Paper 2)

IB-style question — find an unknown constant

Exam Tips

What you'll learn in Topic 1.9

Study resources — 1.9 Binomial theorem

Pascal's triangle & nCr

Binomial expansion

Finding a term

Ready to study Binomial theorem?

Ready to practice?

IB Math AA SL — Binomial theorem

Key concepts in Binomial theorem

🔢 The formulas you're given

✏️ IB-style worked examples

IB-style question — expand a bracket (Paper 1)

IB-style question — a coefficient with the nCr (Paper 2)

IB-style question — find an unknown constant

Exam Tips

What you'll learn in Topic 1.9

Study resources — 1.9 Binomial theorem

Pascal's triangle & nCr

Binomial expansion

Finding a term

Ready to study Binomial theorem?

Ready to practice?