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Quick Summary
The binomial theorem expands (a + b)ⁿ without multiplying it out term by term.
Each term has the form C(n,r) · aⁿ⁻ʳ · bʳ, where C(n,r) = n! / [(n−r)!r!].
Pascal's Triangle gives you the same coefficients — useful for small n without a calculator.
You'll usually only need the first few terms, not the whole expansion.
To find one specific term (e.g. the coefficient of x³), pick out the right value of r using the powers.
Watch signs carefully when b is negative — it flips every other term.
You can be asked to work backwards: given a coefficient, find an unknown constant in the brackets.
1. What Is the Binomial Expansion?
"Binomial" just means two terms — like (a + b). If you wanted to expand
(a + b)² by hand, you'd get a² + 2ab + b². For (a + b)³, you'd get a³ + 3a²b + 3ab² + b³.
Notice a pattern? The powers of a count down, the powers of b count up,
and the numbers in front (the coefficients) follow a set pattern.
Now imagine doing this for (a + b)¹⁰ by hand — multiplying it out ten times would take forever
and you'd almost certainly make an arithmetic slip. The binomial theorem is the shortcut that
gives you every term directly, using a formula for the coefficients instead of grinding through
the multiplication.
In plain words: n is a positive integer (the power you're raising the bracket to).
As you move along the expansion, the power of a drops by 1 each time and the power of b
rises by 1 each time — but they always add up to n. The number in front of each term (the "binomial coefficient")
is calculated using that factorial formula, or read off Pascal's Triangle.
Picture (a + b)ⁿ as choosing, n times, whether you "pick" an a or a "pick" a b, then multiplying your
choices together. C(n,r) simply counts how many different ways you could end up picking b
exactly r times (and a the other n−r times). That's why it's also called "n choose r" — you're literally
choosing r slots out of n to be the b's.
Notation you'll see
C(n,r) might be written a few different ways depending on your textbook or calculator —
they all mean exactly the same thing:
ⁿCr
ₙCr
(n choose r), written as a stacked fraction-like bracket with n on top and r below
Practice Question 1
Write out the general term of the expansion of (a + b)⁷ that contains b⁴. Don't simplify the coefficient yet — just write the term using the formula.
2. Pascal's Triangle
Pascal's Triangle is a visual, no-formula way to get the same binomial coefficients. Each row
corresponds to a power n, and each number in that row corresponds to a C(n,r) value.
The top row (just the number 1) is row 0 — it matches (a+b)⁰.
Every row starts and ends with a 1.
Every other number is the sum of the two numbers diagonally above it.
1
1
1
1
2
1
1
3
3
1
1
4
6
4
1
1
5
10
10
5
1
Rows 0 to 5 — row 4 (1, 4, 6, 4, 1) gives the coefficients for (a+b)⁴
It's like a "family tree" of numbers — each entry is literally born from the two numbers sitting
above it (add them together). That's why the edges are always 1: there's only ever one number
above each edge entry, and 1 + nothing effectively stays 1... more precisely, the very first and
last entry of every row have nothing to "compete" with, so they default to 1.
Pascal's Triangle is great for small values of n in a non-calculator paper.
But for large n (say n = 12 or more), building the whole triangle row by row is slow and easy to
mess up — the C(n,r) formula is faster and more reliable.
Practice Question 2
Use Pascal's Triangle to write out the coefficients needed to expand (a + b)⁶.
3. Expanding a Full Bracket, e.g. (3 − 2x)⁵
In exam questions, a and b are rarely just single letters — they're often numbers and terms with x,
sometimes with a minus sign attached. The method is always the same three steps:
Identify a, b, and n from the bracket.
Substitute them into the binomial formula.
Simplify each term carefully — powers of numbers AND powers of x, and don't forget negative signs.
If b is negative (like −2x), every odd-numbered term in the expansion becomes negative,
because a negative number raised to an odd power stays negative. Students often forget to carry the
negative sign through when squaring or cubing — remember (−2x)² = +4x² but (−2x)³ = −8x³.
Worked Example
Find the first three terms in the expansion of (3 − 2x)⁵, in ascending powers of x.
Step 1 — Identify: a = 3, b = −2x, n = 5
Step 2 — Ascending powers means starting with the constant term (aⁿ), so we use the formula as written:
Find the first three terms, in ascending powers of x, of (5 + 2x)⁹. (You don't need to fully simplify huge numbers — set up each term clearly.)
4. Finding the Coefficient of a Single Term
Most exam questions don't want the whole expansion — they want one specific term,
like "find the coefficient of x⁶". Writing out the entire expansion just to find one term wastes time.
Instead, use the general term formula directly:
General Term Formula
C(n,r) · aⁿ⁻ʳ · bʳ
Work out which value of r gives you the power of x the question is asking for, then plug that single r-value into the formula — no need to build any other term.
How to choose the right r
This depends on where x sits inside the bracket. Use the laws of indices to spot the pattern:
For (a + bx)ⁿ → coefficient of xʳ uses the term aⁿ⁻ʳ(bx)ʳ (r matches the power of x directly)
For (a + bx²)ⁿ → coefficient of xʳ uses a^(n − r/2)(bx²)^(r/2) (halve the target power first)
For (a + b/x)ⁿ → look at how the powers of x will cancel to figure out which r makes that power appear
Think of r as a dial you're turning. Every notch of the dial changes the power of x in that term.
You're not building the whole staircase of terms — you're just figuring out which single step
lands exactly on the power you want, then reading off what's there.
Worked Example
Find the coefficient of x⁶ in the expansion of (3 + 2x)⁸.
Step 1: Here b = 2x, and we want x⁶, so r = 6 (the power of x in bx matches r directly).
Step 2: Total power n = 8, so the power of a is n − r = 8 − 6 = 2.
Step 3: The term is: C(8,6)(3)²(2x)⁶
= 28 × 9 × 64x⁶ = 16 128x⁶
So the coefficient of x⁶ is 16 128
Worked Example — Working Backwards
The coefficient of the x³ term in the expansion of (p + 2x)⁵ is 2000. Find the value of p, given p > 0.
Step 1: We only need the term in x³, so r = 3, n = 5.
Step 3: Set the coefficient equal to 2000: 80p² = 2000
p² = 25 → p = ±5
Since p > 0, reject p = −5. So p = 5.
Always check restrictions like "p > 0" at the very end — squaring loses the sign,
so quadratic-style answers often give you two solutions and only one is valid.
Practice Question 4
Find the coefficient of x⁴ in the expansion of (2 − x)⁷.
5. Applications & Exam Presentation
Binomial expansions in exams are almost always used to expand a bracket of the form (a + b)ⁿ, and you'll typically only need the first three or four terms — not the entire thing.
Ascending vs Descending Powers
Ascending powers of x → start with the constant term aⁿ (the term with x⁰) and work up.
Descending powers of x → start with the term containing the highest power of x. You may need to swap which letter you call "a" and which you call "b" so the standard formula still applies cleanly.
Showing the expansion continues
If you're not writing out the full expansion (which is most of the time), show the reader that more terms exist using one of these:
An ellipsis (…) after your last term — shows the sequence continues exactly
The approximately-equals sign (≈) — shows the terms you've found are an approximation of the full value
Use "=" with "…" when your line is still an exact ongoing expansion. Use "≈" when you're truncating the expansion to only a few terms and treating it as an approximate value (e.g. for estimating a numerical value).
What to Memorise
Binomial Theorem
(a+b)ⁿ = aⁿ + C(n,1)aⁿ⁻¹b + C(n,2)aⁿ⁻²b² + … + bⁿ
Binomial Coefficient
C(n,r) = n! / [(n−r)! · r!]
General Term
C(n,r) · aⁿ⁻ʳ · bʳ — used to find one specific term without expanding everything
Pascal's Triangle Rule
Row starts/ends with 1; every other entry = sum of the two entries diagonally above it
Row Numbering
The first row (just "1") is row 0, matching C(0,0) and (a+b)⁰
Sign Rule
If b is negative, terms alternate sign — even powers of b stay positive, odd powers stay negative
Notation
C(n,r) = ⁿCr = ₙCr — all the same value, different notation styles
Ascending vs Descending
Ascending starts at aⁿ (x⁰ term); descending starts at the highest power of x
Concepts Checklist
Tick off each idea once you're confident you can explain it without looking back.
Exam Tips & Common Traps
Losing negative signs. When b is negative, always put it in brackets, e.g. (−2x), and raise the WHOLE bracket to the power — not just the x. (−2x)² = 4x², not −4x².
Mixing up r with the power of x. r is not always exactly the power of x you want — it depends on the structure of b. If b = x², you need to solve for r first before using it.
Messy working. Binomial expansion arithmetic gets cluttered fast — always keep the binomial coefficient, the power of a, and the power of b on separate visible chunks (in brackets) before multiplying everything together. This is exactly what examiners want to see for method marks, even if your final number is wrong.
Forgetting to reject invalid solutions. "Find an unknown" problems often produce a ± result from squaring. Always check any given restriction (like p > 0) and explicitly reject the invalid root.
Using the wrong starting term for descending powers. If the question asks for descending powers of x, don't just start from an as normal — you likely need to swap a and b (or start counting from the bⁿ end) so your first term genuinely has the highest power of x.
What examiners look for: correct identification of a, b, and n; correct binomial coefficient (from the formula or Pascal's Triangle); correct handling of signs and powers of numbers separately from powers of x; and a fully simplified final coefficient.
Also in the full note
5. Applications & Exam Presentation
Exam Tips & Common Traps
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